The Pennsylvania State University
The Graduate School
Department of Mathematics
A LINEAR SHELL THEORY
BASED ON VARIATIONAL PRINCIPLES
A Thesis in
Mathematics
by
Sheng Zhang
c 2001 Sheng Zhang
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
August 2001
We approve the thesis of Sheng Zhang.
Date of Signature
Douglas N. Arnold
Distinguished Professor of Mathematics
Thesis Adviser
Chair of Committee
M. Carme Calderer
Professor of Mathematics
Chun Liu
Assistant Professor of Mathematics
Eduard S. Ventsel
Professor of Engineering Science and Mechanics
Gary L. Mullen
Professor of Mathematics
Chair, Department of Mathematics
iii
Abstract
Under the guidance of variational principles, we derive a two-dimensional shell
model, which is a close variant of the classical Naghdi model. From the model solution,
approximate stress and displacement fields can be explicitly reconstructed. Convergence
of the approximate fields toward the more accurate three-dimensional elasticity solutions
is proved. Convergence rates are established. Potential superiority of the Naghdi-type
model over the Koiter model is addressed. The condition under which the model might
fail is also discussed.
iv
Table of Contents
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1
Background and motivations . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Organization of this thesis . . . . . . . . . . . . . . . . . . . . . . . .
5
1.3
Principal results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.3.1
Plane strain cylindrical shells . . . . . . . . . . . . . . . . . .
8
1.3.2
Spherical shells . . . . . . . . . . . . . . . . . . . . . . . . . .
10
1.3.3
General shells . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
Chapter 2. Plane strain cylindrical shell model . . . . . . . . . . . . . . . . . . .
19
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.2
Plane strain cylindrical shells . . . . . . . . . . . . . . . . . . . . . .
20
2.2.1
Curvilinear coordinates on a plane domain . . . . . . . . . . .
21
2.2.2
Plane strain elasticity . . . . . . . . . . . . . . . . . . . . . .
22
2.2.3
Plane strain cylindrical shells . . . . . . . . . . . . . . . . . .
25
2.2.4
Rescaled stress and displacement components . . . . . . . . .
29
2.3
The shell model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
2.4
Reconstruction of the stress and displacement fields . . . . . . . . . .
37
2.4.1
37
Reconstruction of the statically admissible stress field . . . .
v
2.4.2
Reconstruction of the kinematically admissible displacement
field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
Constitutive residual . . . . . . . . . . . . . . . . . . . . . . .
41
Justification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
2.5.1
Assumption on the applied forces . . . . . . . . . . . . . . . .
44
2.5.2
An abstract theory . . . . . . . . . . . . . . . . . . . . . . . .
45
2.5.3
Asymptotic behavior of the model solution . . . . . . . . . .
47
2.5.4
Convergence theorem . . . . . . . . . . . . . . . . . . . . . . .
54
Shear dominated shell examples . . . . . . . . . . . . . . . . . . . . .
60
2.6.1
A beam problem . . . . . . . . . . . . . . . . . . . . . . . . .
60
2.6.2
A circular cylindrical shell problem . . . . . . . . . . . . . . .
62
Chapter 3. Analysis of the parameter dependent variational problems . . . . . .
65
2.4.3
2.5
2.6
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
3.2
The parameter dependent problem and its mixed formulation . . . .
65
3.3
Asymptotic behavior of the solution . . . . . . . . . . . . . . . . . .
72
3.3.1
The case of surjective membrane–shear operator . . . . . . .
74
3.3.2
The case of flexural domination . . . . . . . . . . . . . . . . .
78
3.3.3
The case of membrane–shear domination . . . . . . . . . . . .
83
3.4
Parameter-dependent loading functional . . . . . . . . . . . . . . . .
90
3.5
Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
Chapter 4. Three-dimensional shells . . . . . . . . . . . . . . . . . . . . . . . . .
96
4.1
Curvilinear coordinates on a shell . . . . . . . . . . . . . . . . . . . .
96
vi
4.2
Linearized elasticity theory . . . . . . . . . . . . . . . . . . . . . . .
106
4.3
Rescaled components . . . . . . . . . . . . . . . . . . . . . . . . . . .
109
Chapter 5. Spherical shell model . . . . . . . . . . . . . . . . . . . . . . . . . . .
115
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
115
5.2
Three-dimensional spherical shells . . . . . . . . . . . . . . . . . . . .
116
5.3
The spherical shell model . . . . . . . . . . . . . . . . . . . . . . . .
120
5.4
Reconstruction of the admissible stress and displacement fields . . .
124
5.4.1
The admissible stress and displacement fields . . . . . . . . .
125
5.4.2
The constitutive residual . . . . . . . . . . . . . . . . . . . . .
129
Justification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
131
5.5.1
Assumption on the applied forces . . . . . . . . . . . . . . . .
131
5.5.2
Asymptotic behavior of the model solution . . . . . . . . . .
132
5.5.3
Convergence theorem . . . . . . . . . . . . . . . . . . . . . . .
139
5.5.4
About the condition of the convergence theorem . . . . . . .
144
5.5.5
A shell example for which the model might fail . . . . . . . .
146
Chapter 6. General shell theory . . . . . . . . . . . . . . . . . . . . . . . . . . .
148
5.5
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
148
6.2
The shell model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
150
6.3
Reconstruction of the stress field and displacement field . . . . . . .
155
6.3.1
The stress and displacement fields . . . . . . . . . . . . . . .
156
6.3.2
The second membrane stress moment . . . . . . . . . . . . .
160
6.3.3
The integration identity . . . . . . . . . . . . . . . . . . . . .
164
vii
6.3.4
Constitutive residual . . . . . . . . . . . . . . . . . . . . . . .
169
6.3.5
A Korn-type inequality on three-dimensional thin shells . . .
172
Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
174
6.4.1
Assumptions on the loading functions . . . . . . . . . . . . .
174
6.4.2
Classification . . . . . . . . . . . . . . . . . . . . . . . . . . .
175
Flexural shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
178
6.5.1
Asymptotic behavior of the model solution . . . . . . . . . .
179
6.5.2
Convergence theorems . . . . . . . . . . . . . . . . . . . . . .
183
6.5.3
Plate bending . . . . . . . . . . . . . . . . . . . . . . . . . . .
188
Totally clamped elliptic shells . . . . . . . . . . . . . . . . . . . . . .
189
6.6.1
Reformulation of the resultant loading functional . . . . . . .
192
6.6.2
Asymptotic behavior of the model solution . . . . . . . . . .
194
6.6.3
Convergence theorems . . . . . . . . . . . . . . . . . . . . . .
197
6.6.4
Estimates of the K-functional for smooth data . . . . . . . .
198
Membrane–shear shells . . . . . . . . . . . . . . . . . . . . . . . . . .
204
6.7.1
Asymptotic behavior of the model solution . . . . . . . . . .
205
6.7.2
Admissible applied forces . . . . . . . . . . . . . . . . . . . .
213
6.7.3
Convergence theorem . . . . . . . . . . . . . . . . . . . . . . .
216
Chapter 7. Discussions and justifications of other linear shell models . . . . . . .
222
6.4
6.5
6.6
6.7
7.1
Negligibility of the higher order term in the loading functional . . . .
223
7.2
The Naghdi model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
226
7.3
The Koiter model and the Budianski–Sanders model . . . . . . . . .
227
viii
7.4
The limiting models . . . . . . . . . . . . . . . . . . . . . . . . . . .
230
7.5
About the loading assumption . . . . . . . . . . . . . . . . . . . . . .
232
7.6
Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . .
234
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
235
ix
List of Figures
2.1
A cylindrical shell and its cross section . . . . . . . . . . . . . . . . . . .
27
2.2
Deformations of a cylindrical shell . . . . . . . . . . . . . . . . . . . . .
50
2.3
Shear dominated deformation of a beam . . . . . . . . . . . . . . . . . .
60
2.4
Shear dominated deformation of a circular cylinder . . . . . . . . . . . .
62
4.1
A shell and its coordinate domain . . . . . . . . . . . . . . . . . . . . . .
105
x
Acknowledgments
I am most grateful and indebted to my thesis advisor, Prof. Douglas N. Arnold,
for his advice that led to my selection of the project, his penetrating vision that guided me
moving forward, his enthusiasm toward mathematics that made me feel the mathematical
research somewhat enjoyable, and for his patience, encouragement, and support.
I wish to thank Professors M. Carme Calderer, Chun Liu, and Eduard S. Ventsel
who kindly agreed to serve on my thesis committee.
During the writing of this thesis, I was supported by the Pritchard Dissertation
Fellowship, which is gratefully acknowledged.
1
Chapter 1
Introduction
1.1
Background and motivations
A shell is a three-dimensional elastic body occupying a thin neighborhood of a
two-dimensional manifold, which resists deformation owing to the material of which it
is made, its shape, and boundary conditions. It is extremely important in structural
mechanics and engineering because a well-designed shell can sustain large loads with
remarkably little material. For example, before collapsing, a totally clamped spherical
shell of thickness 2 can hold a strain energy of O(−1/3 ) times that which can be
tolerated by a flat plate of the same thickness (see page 202). For this reason, shells are
a favored structural element in both natural and man-made constructions. While elastic
shells can exhibit great strength, their behaviors can also be very difficult to predict,
and they can fail in a catastrophic fashion.
Although the deformation of a shell arising in response to given loads and boundary conditions can be accurately captured by solving the three-dimensional elasticity
equations, shell theory attempts to provide a two-dimensional representation of the intrinsically three-dimensional phenomenon [34]. There are two reasons to derive a lower
dimensional model. One is its simpler mathematical structure. For example, the existence, regularity, bifurcation, and global analysis are by now on firm mathematical
2
grounds for non-linear elastic rods [18]. In contrast, the mathematical theory for nonlinear three-dimensional elasticity is much less developed. Another motivation is for
numerical simulation. An accurate, fully three-dimensional, simulation of a very thin
body is beyond the power of even the most powerful computers and computational
techniques. Furthermore, the standard methods of numerical approximation of threedimensional elastic bodies fail for bodies which are thin in some direction, unless the
behavior is resolved in that direction. Thus the need for two-dimensional shell models
[5].
Beginning in the late nineteenth century, and especially during the past few
decades, there have been intense efforts to derive an accurate dimensionally reduced
mathematical theory of shells. Despite much progress, the development of a satisfactory mathematical theory of elastic shells is far from complete. The methodologies for
deriving shell models from three-dimensional continuum theories are still being developed, and the relation between different approaches, are not clear. Controversial issues
abound. The extremely important question of deriving rigorous mathematical theory
relating shell models to more exact three-dimensional models is wide open. A thorough
analysis of the mathematical models derived and a rigorous definition of their ranges of
applicability is mostly lacking.
There is a huge literature devoted to dimensional reduction in elasticity theory.
Several classical approaches are employed in investigations. One approach starts with a
priori assumptions on the displacement and stress fields based on mechanical considerations, such as the Kirchhoff–Love assumption on the displacements and the kinetic assumption on the stress fields that assumes both the transverse shear and normal stresses
3
are negligible. This approach leads to the biharmonic plate bending model, Koiter shell
model, flexural shell model, and many others. Models derived in this way have proved
successful in practice, but this approach does not seem to lend itself naturally to an error
analysis [2].
Another approach is through a formal asymptotic analysis in which the thickness
of the elastic body is viewed as a small parameter. By expanding the three-dimensional
elasticity equation with respect to the thickness, the leading terms in the expansion
are used to define lower dimensional models. This approach leads to limiting models
describing the zero thickness limit situation, among which are the limiting flexural and
membrane models, depending on ad hoc assumptions on the applied forces, the shell
geometry, and boundary conditions. These asymptotic methods only lead to the limiting
models. It does not seem to be possible to derive the better Koiter and Naghdi models
by this approach. (Taking more terms in the asymptotic expansion does not lead to a
dimensionally reduced model.) See [18] for a comprehensive treatment of this approach.
A third approach is by variational methods. Solution of the three-dimensional
equation can be characterized by variational principles or weak formulations. An approximation is determined by restricting to a trial space of functions that are finite
dimensional with respect to the transverse variable. By its very nature, this approach
leads to models that yield a displacement field or a stress field determined by finitely
many functions of two variables. Thus the dimension is reduced. In this approach, the
two energies principle, or the Prager–Synge theorem [54], plays a fundamental role in
the model validation. To apply the two energies principle, we must have a statically
admissible stress field and a kinematically admissible displacement field. The latter is
4
usually easy to come by, but the former might be formidable to obtain. The two energies
principle is particularly suited to analyzing complementary energy variational models,
which automatically yield statically admissible stress fields.
The application of the two energies principle to justify plate theory was initiated
in the pioneering work of Morgenstern [47], where it was used to prove the convergence
of the biharmonic model of plate bending when the thickness tends to zero. The statically admissible stress field and the kinematically admissible displacement field were
constructed based on the biharmonic solution in an ad hoc fashion, as needed for the
convergence proof. Following this work, substantial efforts have been made to modify
the justification of the classical plate bending models, see [48], [51], and [57]. In the
same spirit, Gol’denveizer [29], Sensenig [56], Koiter [33], Mathúna [46], and many others considered the error estimates for shell theories. In these latter works, the stress
fields constructed from the model solutions were only approximately admissible, and the
justifications obtained were largely formal.
Due to the formidable difficulty involved in the construction of an admissible stress
field based on the solution of a known model, it seems a better choice to reconsider the
derivation of the model while keeping in mind the construction of the statically admissible
stress field as a primary goal. Based on the Hellinger–Reissner variational formulations
of the three-dimensional elasticity, a systematic procedure of dimensional reduction for
plate problems was developed in [2]. In this approach both the stress and displacement
fields were restricted to subspaces in which functions depend on the transverse coordinate
polynomially. The derivation based on the second Hellinger–Reissner principle not only
led to the well known Reissner–Mindlin plate model but also furnished an admissible
5
stress field and so naturally led to a rigorous justification of the model by the two
energies principle. This approach is not easily extensible to shell problems. Due to the
curved shape of a shell, if this approach were carried over and the subspaces were chosen
to be composed of functions depending on the transverse coordinate polynomially, the
polynomials would be of conspicuously higher order. The resulting model would contain
so many unknowns that it would be nearly as untractable as the three-dimensional model.
In this work we derive and rigorously justify a two-dimensional shell model guided
by the variational principles.
1.2
Organization of this thesis
We consider the modeling of the deformation arising in response to applied forces
and boundary conditions of an arbitrary thin curved shell, which is made of isotropic
and homogeneous elastic material whose Lamé coefficients are λ and µ. The shell is
clamped on a part of its lateral face and is loaded by a surface force on the remaining
part of the lateral face. The shell is subjected to surface tractions on the upper and
lower surfaces and loaded by a body force. We take the three-dimensional linearized
elasticity equation as the supermodel and approximate it by a two-dimensional model.
The lower dimensional model will be justified by proving convergence and establishing the
convergence rate of the model solution to the solution of the three-dimensional elasticity
equation in the relative energy norm under some assumptions on the applied forces.
Conditions under which the model might fail will be discussed. The two energies principle
supplies important guidance for the construction of the model.
6
Throughout the thesis, Greek subscripts and superscripts, except , which is reserved for the half-thickness of the shell, always take their values in {1, 2}, while Latin
scripts always belong to the set {1, 2, 3}. Summation convention with respect to repeated
superscripts and subscripts will be used together with these rules. We usually use lower
case Latin letters with an undertilde, as v , to denote two-dimensional vectors. Lower case
∼
Greek letters with double undertildes denote two-dimensional second order tensors, as
σ . However, the fundamental forms on the shell middle surface will be denoted by lower
∼
case Latin letters. We use boldface Latin letters to denote three-dimensional vectors and
boldface Greek letters second order three-dimensional tensors. Vectors and tensors will
be given in terms of their covariant components, or contravariant components, or mixed
components.
The notation P ' Q means there exist constants C1 and C2 independent of , P ,
and Q such that C1 P ≤ Q ≤ C2 P . The notation P . Q means there exists a constant
C independent of , P , and Q such that P ≤ CQ.
Chapters 2–6 form the main body of the thesis, with Chapters 2 and 5 treating
two special kinds of shells, namely, the plane strain cylindrical shells and spherical shells,
respectively; Chapter 6 treating general shells; and Chapters 3 and 4 containing results
needed for the analysis. The reason we treat cylindrical and spherical shells separately is
that for these special shell problems, we can construct statically admissible stress fields
and kinematically admissible displacement fields, so that we can use the two energies
principle to justify the models by bounding the constitutive residuals. As a consequence,
stronger convergence results can be obtained for these cases. These two special shells
provide examples for all kinds of shells as classified in Section 3.5. For the general
7
shells treated in Chapter 6, precisely admissible stress fields are no longer possible to
construct. The derivation yields an almost admissible stress field with small residuals
in the equilibrium equation and lateral traction boundary condition. The two energies
principle can not be directly used to justify the model. As an alternative, we establish
an integration identity to incorporate all these residuals so that we can bound the model
error by estimating these residuals.
All the models we derive can be written in variational forms, in which the flexural
energy, membrane energy, and shear energy are combined together in the total strain
energy. Contributions of the component energies are weighted by factors that depend
on . Chapter 3 is devoted to the mathematical analysis of such -dependent problems
on an abstract level. In this chapter, we classify the model and analyze the asymptotic
behavior of the model solution when the shell thickness approaches zero. The range
of applicability of the derived model will also be discussed on the abstract level. The
rigorous validation of the shell model crucially hinges on these analyses.
In Chapter 4, we briefly summarize the three-dimensional linearized elasticity theory expressed in the curvilinear coordinates on a thin shell. We also derive some formulas
that can substantially simplify calculations. Finally, in Chapter 7, we will discuss the
relations between our theory and other existing shell theories. In the remainder of this
introduction, we will describe the principal results of the following chapters.
1.3
Principal results
In this section we summerize the key results of Chapters 2, 5, and 6.
8
1.3.1
Plane strain cylindrical shells
In Chapter 2 we consider the simplest case of plane strain cylindrical shells. In
this case, the three-dimensional problem is essentially a two-dimensional problem defined
on a cross-section, so the dimensionally reduced model should be one-dimensional. We
assume that the cylindrical shell is clamped on the two lateral sides, subjected to surface
forces on the upper and lower surfaces, and loaded by a body force.
Let the middle curve of a cross-section of the cylindrical shell be parameterized
by its arc length variable x ∈ [0, L]. Our model can be written as a one-dimensional
variational problem defined on the space H = [H01 (0, L)]3 . The solution of the model is
composed of three single variable functions that approximately describe the shell deformation arising in response to the applied forces and boundary conditions. We introduce
the following operators. For any (θ, u, w) ∈ H, we define
γ(u, w) = ∂u − bw,
ρ(θ, u, w) = ∂θ + b(∂u − bw),
τ (θ, u, w) = θ + ∂w + bu,
which give the membrane strain, flexural strain, and transverse shear strain engendered
by the displacement functions (θ, u, w). Here b is the curvature of the middle curve,
which is a function of the arc length parameter, and ∂ = d/dx.
The model (cf., (2.3.2) below) reads: Find (θ , u , w ) ∈ H, such that
Z L
1 2
?
(2µ + λ )
ρ(θ , u , w )ρ(φ, y, z)dx
3
0
9
+ (2µ + λ? )
Z L
5
γ(u , w )γ(y, z)dx + µ
6
0
Z L
τ (θ , u , w )τ (φ, y, z)dx
0
= hf 0 + 2 f 1 , (φ, y, z)i, ∀(φ, y, z) ∈ H,
in which
λ? =
2µλ
2µ + λ
and the loading functional f 0 +2 f 1 is explicitly expressible in terms of the applied force
functions, cf., (2.3.3), (2.3.4). We show that the solution of this one-dimensional model
uniquely exists. The three single variable functions θ , u , and w that comprise the
model solution describe the rotations of straight fibers normal to the middle curve, the
tangential displacements, and transverse displacements of points on the middle curve,
respectively.
In addition to the model, in Section 2.4 we give formulae to reconstruct a tensor
field σ and a vector field v from the model solution on the shell cross-section, see
∼
∼
equations (2.4.1), (2.4.3), (2.4.7), and (2.4.8). The model and reconstruction formulae
are designed to have the following properties:
(1) σ is a statically admissible stress field (see Section 2.4.1).
∼
(2) v is a kinematically admissible displacement field (see Section 2.4.2).
∼
(3) The terms of leading order in in the constitutive residual Aαβλγ σ λγ −χαβ (v )
∼
vanish, so the constitutive residual may be shown to be small as → 0 (see Section 2.4.3).
10
This allows a bound on the errors of σ and v by the two energies principle. Under
∼
∼
the loading assumptions (2.3.6) and (2.5.1), we prove the inequality
kσ ∗ − σ kE + kχ (v ∗ ) − χ (v )kE ∼
∼ ∼
∼
kχ (v )kE ∼ ∼
. 1/2 ,
∼ ∼
in which σ ∗ is the stress field and v ∗ the displacement field arising in the shell determined
∼
∼
from the two-dimensional elasticity equations. The norm k · kE is the energy norm of
the strain or stress field.
1.3.2
Spherical shells
For spherical shells, we derive the model by a similar method. We assume the
middle surface of the shell is a portion of a sphere of radius R. The shell is clamped
on a part of its lateral face, and subjected to surface force on the remaining part of the
lateral face whose density is linearly dependent on the transverse variable. The shell
is subjected to surface forces on the upper and lower surfaces, and loaded by a body
force whose density is assumed to be constant in the transverse coordinate. The middle
surface is parameterized by a mapping from a domain ω ⊂ R2 onto it. The boundary
∂ω is divided as ∂ω = ∂D ω ∪ ∂T ω giving the clamping and traction parts of the the
lateral face of the shell. The model is a two-dimensional variational problem defined on
1 (ω). The solution of the model is composed of five
the space H = H 1D (ω) × H 1D (ω) × HD
∼
∼
two variable functions that can approximately describe the shell displacement arising in
11
response to the applied loads and boundary conditions. For ( θ , u , w) ∈ H, we define
∼ ∼
1
γαβ (u , w) = (uα|β + uβ|α ) − baαβ w,
∼
2
1
ραβ ( θ ) = (θα|β + θβ|α ), τβ ( θ , u , w) = θβ + ∂β w + buβ ,
∼
∼ ∼
2
which give the membrane, flexural, and transverse shear strains engendered by the displacement functions ( θ , u , w). Here, aαβ is the covariant metric tensor and b = −1/R is
∼ ∼
the curvature of the middle surface. The model (cf., (5.3.2)) reads: Find ( θ , u , w ) ∈
∼ ∼
H, such that
Z
√
1 2
aαβλγ ρλγ ( θ )ραβ (φ ) adx
∼
∼
∼
3
ω
Z
Z
√
√
5
αβλγ
+
a
γλγ (u , w )γαβ (v , z) adx + µ
aαβ τβ ( θ , u , w )τα (φ , v , z) adx
∼
∼
∼
∼
∼
∼
∼
∼
6 ω
ω
= hf 0 + 2 f 1 , (φ , y , z)i, ∀ (φ , y , z) ∈ H
∼ ∼
∼ ∼
where aαβ is the contravariant metric tensor of the middle surface and
aαβλγ = 2µaαλ aβγ + λ? aαβ aλγ
(1.3.1)
is the two-dimensional elasticity tensor of the shell. The resultant loading functional
f 0 + 2 f 1 can be explicitly expressed in terms of the applied force functions, cf., (5.3.3),
(5.3.4). This model has a unique solution if the resultant loading functional is in the dual
space of H. This condition is satisfied if the applied force functions satisfy the condition
(5.3.6). The unique solution ( θ , u , w ) describes the normal straight fiber rotations,
∼ ∼
12
middle surface tangential displacement and transverse displacement, respectively. A
statically admissible stress field and a kinematically admissible displacement field can
be reconstructed from the model solution. We prove the convergence and establish the
convergence rate of the model solution to the three-dimensional solution by estimating
the constitutive residual.
1.3.3
General shells
For a general shell, except for some smoothness requirements, we do not impose
any restriction on the geometry of the shell middle surface or the shape of its lateral
boundary. The shell is assumed to be clamped on a part of its lateral surface and loaded
by a surface force on the remaining part. The shell is subjected to surface forces on the
upper and lower surfaces, and loaded by a body force.
The model is constructed in the vein of the model constructions for the special
shells in the Chapters 2 and 5. The main difficulty to overcome is that our model
derivation does not yield a statically admissible stress field. Therefore, the two energies
principle can not be directly used to justify the model. Even so, we can reconstruct a
stress field that is almost admissible with small residuals in the equilibrium equation
and lateral traction boundary condition. And we will establish an integration identity
(6.3.17) to incorporate the equilibrium residual and the lateral traction boundary condition residual. This identity plays the role of the two energies principle in the general
shell theory.
Let the middle surface of the shell be parameterized by a mapping from the domain
ω ⊂ R2 onto it. Corresponding to the clamping and traction parts of the lateral face,
13
the boundary of ω is divided as ∂ω = ∂D ω ∪ ∂T ω. In this curvilinear coordinates, the
fundamental forms on the shell middle surface are denoted by aαβ , bαβ , and cαβ . The
mixed curvature tensor is denoted by bα
β . The model is a two-dimensional variational
1 (ω). The solution of the
problem defined on the space H = H 1D (ω) × H 1D (ω) × HD
∼
∼
model is composed of five two variable functions that can approximately describe the
shell displacement arising in response to the applied loads and boundary conditions. For
( θ , u , w) ∈ H, we define the following two-dimensional tensors.
∼ ∼
1
γαβ (u , w) = (uα|β + uβ|α ) − bαβ w,
∼
2
1
1
u
+ bλ
ραβ ( θ , u , w) = (θα|β + θβ|α ) + (bλ
α uβ|λ ) − cαβ w,
∼ ∼
2
2 β α|λ
τβ ( θ , u , w) = bλ
β uλ + θβ + ∂β w.
∼ ∼
These two-dimensional tensor- and vector-valued functions give the membrane strain,
flexural strain, and transverse shear strain engendered by the displacement functions
( θ , u , w), respectively. The model (cf., (6.2.4)) reads: Find ( θ , u , w ) ∈ H, such that
∼ ∼
∼ ∼
Z
√
1 2
aαβλγ ρλγ ( θ , u , w )ραβ (φ , y , z) adx
∼
∼
∼
∼
∼
3
ω
Z
Z
√
√
5
αβλγ
+
a
γλγ (u , w )γαβ (y , z) adx + µ
aαβ τβ ( θ , u , w )τα (φ , y , z) adx
∼ ∼
∼ ∼
∼
∼
∼ 6
∼
ω
ω
= hf 0 + 2 f 1 , (φ , y , z)i, ∀(φ , y , z) ∈ H,
∼ ∼
∼ ∼
in which the fourth order two-dimensional contravariant tensor aαβλγ is the elastic tensor
of the shell, defined by the formula (1.3.1). The resultant loading functional f 0 + 2 f 1
14
can be explicitly expressed in terms of the applied force functions, cf., (6.2.5), (6.2.6).
This model has a unique solution if the resultant loading functional is in the dual space of
H, a condition that can be easily satisfied. From the model solution, we can reconstruct
a stress field σ by explicitly giving its contravariant components. By a correction to
the transverse deflection, we can define a displacement field v by giving its covariant
components. Under some conditions, we will prove the convergence of both σ and v
to the stress and displacement fields determined from the three-dimensional elasticity
equation by using the aforementioned identity and bounding the constitutive residual,
equilibrium equation residual and lateral traction boundary condition residual.
The model is a close variant of the classical Naghdi shell model. This model differs
from the generally accepted Naghdi model in three ways. First, the resultant loading
functional has a somewhat more involved form. Second, the coefficient of the shear
term is 5/6 rather than the usual value 1. The “best” choice for this coefficient seems an
unresolved issue for shells. When the shell is flat, the model degenerates to the Reissner–
Mindlin plate bending and stretching models for which the corresponding value 5/6 is
often accepted as the best, see [55] and [2]. The third, and most significant, difference
is in the expression of the flexural strain ραβ . The relation between our definition and
that of Naghdi’s (ρN
αβ ) is
γ
λ
ραβ = ρN
αβ + bα γλβ + bβ γγα .
We will see that the change of the flexural strain expression appears to be necessary to
make the constitutive residual small in some cases (see Remark 6.3.2). In most cases, this
15
difference does not affect the convergence of the model solution to the three-dimensional
solution.
When the general shell model is applied to spherical shells, we obtain a spherical
shell model slightly different from what we derived in Chapter 5 both in the form of the
flexural strain and in the resultant loading functional. The convergence properties of
these two spherical shell models are the same. What we can learn from this discrepancy
is that the model can be changed, but the resultant loading functional must be changed
accordingly, otherwise a variation in the form of a model might lead to divergence.
To prove convergence, we need to make an assumption on the dependence of the
applied force functions on the shell thickness. We will assume that all the applied force
functions that are explicitly involved in the resultant loading functional are independent
of . Under this assumption, by properly defining function spaces and operators, the
shell models can be abstracted to the variational problem:
2 (Au, Av)U + (Bu, Bv)V = hf0 + 2 f1 , viH ∗ ×H ,
(1.3.2)
u ∈ H,
∀ v ∈ H,
where H, U , and V are Hilbert spaces. The functionals f0 and f1 are independent of .
The linear bounded operators A and B are from H to U and V respectively, with the
property
kAuk2U + kBuk2V ' kuk2H
∀ u ∈ H.
We can assume that the range W of the operator B is dense in V , and equip W with
a norm to make it a Hilbert space. For the plane strain cylindrical shell model, we
16
can prove that the operator B has closed range. This special property substantially
simplifies the analysis of behavior of the model solution and significantly strengthes the
convergence results.
The asymptotic behavior of the solution of this abstract problem is mostly determined by the leading term f0 in the loading functional. We classify the problem as a
flexural shell problem if f0 |ker B 6= 0. For flexural shells, after scaling the applied forces,
the model can be viewed as the penalization of the limiting flexural shell model, which is
constrained on ker B and independent of . The behavior of the model solution and its
convergence property to the three-dimensional solution crucially hinge on the regularity
of the Lagrange multiplier ξ 0 of this constrained limiting problem. Without any extra
assumption, we have ξ 0 ∈ W ∗ and the convergence
kσ ∗ − σkE + kχ(v ∗ ) − χ(v)kE = 0,
kχ(v)kE →0
lim
(1.3.3)
in which σ ∗ is the stress field and v ∗ is the displacement field determined from the threedimensional elasticity. The norm is the energy norm and χ(v) is the three-dimensional
strain field engendered by the displacement v.
The convergence rate essentially depends on the position of ξ 0 between V ∗ and
W ∗ . Under the assumption (6.5.14), we can prove the inequality
kσ ∗ − σkE + kχ(v ∗ ) − χ(v)kE . θ ,
kχ(v)kE (1.3.4)
17
in which θ ∈ [0, 1]. Note that the case of θ = 0 corresponds to the situation that ξ 0 is
only in W ∗ . The previous convergence result can not be deduced from this result on the
convergence rate.
If f0 |ker B = 0, by the closed range theorem in functional analysis, there exists a
unique ζ∗0 ∈ W ∗ such that the leading term of the resultant loading functional can be
reformulated as
hf0 , viH ∗ ×H = hζ∗0 , BviW ∗ ×W ∀ v ∈ H.
If we only have ζ∗0 ∈ W ∗ , we can not prove any convergence. Very likely, the model
diverges in the energy norm in this case. If ζ∗0 ∈ V ∗ , the abstract problem will be called
a membrane–shear problem. This condition is a necessary requirement for us to prove the
convergence of the model solution to the three-dimensional solution. Under this condition
and the assumption that the applied forces are admissible (the admissible assumption on
the applied forces is not needed for spherical shells), we can prove a convergence of the
form (1.3.3). The convergence rate is determined by where ζ 0 , the Riesz representation
of ζ∗0 in V , stands between W and V . For a totally clamped elliptic shell, which is a
special example of membrane–shear shells, under some smoothness assumption on the
shell data in the Sobolev sense, we prove the convergence rate
kσ ∗ − σkE + kχ(v ∗ ) − χ(v)kE . 1/6 .
kχ(v)kE If the odd part of the tangential surface forces vanishes, the convergence rate O(1/5 )
can be proved.
18
The condition ζ∗0 ∈ V ∗ is essentially equivalent to the existence condition for a
solution of the “generalized membrane” shell model defined in [18]. This condition is
trivially satisfied for shear dominated plane strain cylindrical shells. For shear dominated
plate bending, the condition is satisfied as long as the loading function belongs to L2 .
The condition is acceptable for stiff parabolic shells and stiff hyperbolic shells. It can
be satisfied for a totally clamped elliptic shell if the shell data are fairly smooth in the
Sobolev sense. But it imposes a stringent restriction for a partially clamped elliptic shell,
in which case even if the shell data are infinitely smooth, the condition might not be
satisfied. If the condition is not satisfied, although the model solution always exists, a
rigorous relation to the three-dimensional solution is completely lacking.
To reveal the potential advantages of using the Naghdi-type model, we need a
different assumption on the applied force functions. Specifically, we assume that the odd
part of the applied surface forces has a bigger magnitude than what usually assumed.
Under this assumption and in the convergent case of membrane–shear shells, the model
solution violates the Kirchhoff–Love hypothesis on which the Koiter shell model were
based. Therefore it can not converge.
Finally, in the last chapter, we give justifications for other linear shell models
based on the convergence theorems proved for the general shell model, and we will show
that under the usual loading assumption, the differences between our model and other
models are not significant.
For lack of space, we excluded the model derivations. We will directly present the
models and address the more important issue of rigorous justifications.
19
Chapter 2
Plane strain cylindrical shell model
2.1
Introduction
The shell problem of this chapter is a special example of general shells. The
mathematical structure of the derived model is much simpler and we can get much
stronger results on the model convergence. Although the problem is simple, it reveals
our basic strategy to tackle the general problem.
We consider a 3D elastic body that is an infinitely long cylinder whose cross
section is a curvilinear thin rectangle. The body is clamped on the two lateral sides and
subjected to surface traction forces on the upper and lower surfaces and loaded by a
body force. The applied forces are assumed to be in the sectional plane. Under these
assumptions, the elasticity problem is a plane strain problem and can be fully described
by a 2D problem defined on a cross section. We assume that the width 2 of the sectional
curvilinear rectangle is much smaller than its length, so the cylinder is a thin shell.
When the shell is thin, it is reasonable to approximately reduce the 2D elasticity
problem to a 1D problem defined on the middle curve of a cross section. A system of
ordinary differential equations defined on the middle curve that can effectively capture
the displacement and stress of the shell arising in response to the applied forces and
boundary conditions will be the desired shell model.
20
The model, which is a close variant of the Naghdi shell model, is constructed
under the guidance of the two energies principle. The plane strain elasticity problem
and the two energies principle will be briefly described in section 2.2. The model will be
presented and the existence and uniqueness of its solution will be proved in Section 2.3.
We reconstruct the admissible stress and displacement fields from the model solution
and compute the constitutive residual in Section 2.4. In Section 2.5, we analyze the
asymptotic behavior of the model solution and prove the convergence theorem.
Our conclusion is that when the limiting flexural model has a nonzero solution,
our model solution converges to the exact solution at the rate of 1/2 in the relative
energy norm. In this case, the model is just as good as the limiting flexural model and
Koiter model. When the solution of the limiting flexural model is zero, our model gives
a solution that can capture the membrane and shear deformations, and the convergence
rate in the relative energy norm is still 1/2 . The non-vanishing transverse shear deformation violates the Kirchhoff–Love hypothesis in this case. Finally, to emphasize the
necessity of using the Naghdi-type model in some cases, we give two examples in which
the deformations are shear-dominated, which can be very well captured by our model,
but is totally missed by the limiting flexural model and the Koiter model.
2.2
Plane strain cylindrical shells
Since the cross section of the cylindrical shell is a curvilinear rectangle, it is
advantageous to work with curvilinear coordinates. In this section we briefly describe
the plane strain elasticity theory in curvilinear coordinates for a cylindrical shell.
21
2.2.1
Curvilinear coordinates on a plane domain
Let ω ⊂ R2 be an open domain, and (x1 , x2 ) be the Cartesian coordinates of a
generic point in it. Let Φ : ω̄ → R2 be an injective mapping. We assume that Ω = Φ(ω)
is a connected open domain and ∂Ω = Φ(∂ω). The pair of numbers (x1 , x2 ) then
furnish the curvilinear coordinates on Ω. At any point along the coordinate lines, the
tangential vectors g α = ∂Φ/∂xα form the covariant basis. The covariant components
gαβ of the metric tensor are given by gαβ = g α · g β . The contravariant basis vectors are
determined by the relation g α · g β = δβα . The contravariant components of the metric
tensor are g αβ = g α · g β . Note that g αλ gλβ = δβα . The Christoffel symbols are defined
λ
by Γ∗λ
αβ = g · ∂β g α .
Any vector field v defined on Ω can be expressed in terms of its covariant com∼
ponents vα or contravariant components v α by v = vα g α = v α g α . Any second-order
∼
tensor field σ can be expressed in terms of its contravariant σ αβ or covariant components
∼
σαβ by σ = σ αβ g α ⊗ g β = σαβ g α ⊗ g β .
∼
The covariant derivative, a second order tensor field, of a vector field v is defined
∼
in terms of covariant components by
vαkβ = ∂β vα − Γ∗λ
αβ vλ ,
(2.2.1)
which is the gradient of the vector field.
The covariant derivative of a tensor field with contravariant components σ αβ is
defined by
∗β
γβ + Γ σ αγ ,
σ αβ kλ = ∂λ σ αβ + Γ∗α
λγ σ
λγ
(2.2.2)
22
which are mixed components of a third order tensor field. The row divergence of the
tensor field σ αβ is a vector field resulting from a contraction of this third order tensor,
∗β
βγ + Γ σ αγ .
div σ = σ αβ kβ = ∂β σ αβ + Γ∗α
βγ σ
βγ
∼
(2.2.3)
The components of a vector or tensor field defined over Ω can be viewed as functions
defined on the coordinate domain ω.
2.2.2
Plane strain elasticity
Let an infinitely long cylindrical elastic body occupying the 3D domain Ω ×
(−∞, ∞) ⊂ R3 be clamped on a part of its surface ∂D Ω × (−∞, ∞). On the remaining
part of the surface ∂T Ω × (−∞, ∞), the body is subjected to the surface traction force
whose density p is in the Ω-plane and independent of the longitudinal direction. If the
∼
applied body force q is also assumed to be in the Ω-plane and independent of the longi∼
tudinal direction, the displacement of the body arising in response to the applied forces
and clamping boundary condition will be in the plane of Ω and constant in the longitudinal direction. The displacement can be represented by a 2D vector field v and the strain
∼
by a 2D tensor field χ defined on Ω. The stress field can also be treated as a 2D tensor
∼
field σ that is composed of the in-plane components. Although the stress component
∼
in the direction normal to the Ω-plane does not vanish, it is totally determined by the
in-plane stress components.
23
The following five equations (2.2.4–2.2.8) constitute the theory of plane strain
elasticity. The theory includes the geometric equation
1
χαβ (v ) = (vαkβ + vβkα )
∼
2
(2.2.4)
and the constitutive equation
σ αβ = C αβλγ χλγ ,
or χαβ = Aαβλγ σ λγ ,
(2.2.5)
where the fourth order tensors C αβλγ and Aαβλγ are the plane strain elasticity tensor
and the compliance tensor respectively, given by
C αβλγ = 2µg αλ g βγ + λg αβ g λγ
and
Aαβλγ =
1
λ
[gαλ gβγ −
g g ],
2µ
2(µ + λ) αβ λγ
in which λ and µ are the Lamé coefficients of the elastic material comprising the cylinder.
To describe the equilibrium equation and boundary conditions, we need more notations.
We denote the unit outward normal on the boundary ∂Ω by n = nα g α . Let the surface
∼
force density be p = pα g α , and body force density be q = q α g α . With these notations,
∼
∼
the equilibrium equation can be written as
σ αβ kβ + q α = 0.
(2.2.6)
24
On the part of the domain boundary ∂T Ω, the surface force condition can be expressed
as
σ αβ nβ = pα .
(2.2.7)
On ∂D Ω, the body is clamped, so the condition is
vα = 0.
(2.2.8)
According to the linearized elasticity theory, the system of equations (2.2.4),
(2.2.5), (2.2.6) together with the boundary conditions (2.2.7) and (2.2.8) uniquely de∗ of the displacement field of the elastic body
termine the covariant components vα
arising in response to the applied forces and the prescribed clamping boundary condition. The stress distribution σ ∗ is determined by giving its contravariant components
∼
σ ∗αβ = C αβλγ χλγ (v ∗ ).
∼
The weak formulation of the plane strain elasticity equation is
Z
Ω
C αβλγ χλγ (v )χαβ (u ) =
∼
Z
∼
Ω
q α uα +
Z
∂T Ω
pα u α ,
(2.2.9)
v ∈ H 1D (ω) ∀ u ∈ H 1D (ω),
∼
∼
∼
∼
in which H 1D (ω) is the space of vector-valued functions that are square integrable and
∼
have square integrable first derivatives, and vanish on ∂D ω. It is clear that if q α is in
1/2
the dual space of H 1D (ω), and pα is in the dual space of the trace space H 00 (∂T ω), the
∼
∼
variational problem has a unique solution v ∗ ∈ H 1D (ω).
∼
∼
25
A symmetric tensor field σ is called a statically admissible stress field if it satisfies
∼
both the equilibrium equation (2.2.6) and the traction boundary condition (2.2.7). A
vector field v ∈ H 1 (ω) is called a kinematically admissible displacement field, if it
∼
∼
satisfies the clamping boundary condition (2.2.8). For a statically admissible field σ and
∼
a kinematically admissible field v , the following integration identity holds:
∼
Z
Ω
Aαβλγ (σ λγ − σ ∗λγ )(σ αβ − σ ∗αβ )
Z
+
Ω
C αβλγ [χλγ (v ) − χλγ (v ∗ )][χαβ (v ) − χαβ (v ∗ )]
∼
∼
∼
∼
Z
=
[σ αβ − C αβλγ χλγ (v )][Aαβλγ σ λγ − χαβ (v )]. (2.2.10)
∼
∼
Ω
This is the two energies principle, from which the minimum complementary energy
principle and minimum potential energy principle easily follow. If we somehow obtain
an approximate admissible stress field σ and an approximate admissible displacement
∼
field v , then the two energies principle gives an a posteriori bound for the accuracies of
∼
σ and v in the energy norm by the norm of the residual of the constitutive equation.
∼
∼
For the plane strain cylindrical shell problem, this identity will direct us to a model, and
enable us to justify it.
2.2.3
Plane strain cylindrical shells
A plane strain cylindrical shell problem is a special plane strain elasticity problem,
in which the cross section of the cylinder is a thin curvilinear rectangle. For simplicity,
26
we assume that it is clamped on the two lateral sides and subjected to surface forces on
its upper and lower surfaces, and loaded by a body force.
Let the middle curve S ⊂ R2 of the cross section be parameterized by its arc
length through the mapping φ, i.e.,
S = {φ(x)|x ∈ [0, L]}.
With this parameterization, the tangent vector a1 = ∂φ/∂x is a unit vector at any
point on S. At each point on S, we define the unit vector a2 that is orthogonal to the
curve and lies on the same side of the curve for all points.
The cross section Ω of the cylindrical shell, with middle curve S and thickness
2 , occupies the region in R2 that is the image of the thin rectangle ω = [0, L] × [− , ]
through the mapping
Φ(x, t) = φ(x) + ta2 ,
x ∈ [0, L], t ∈ [− , ].
We assume that is small enough so that Φ is injective. The pair of numbers (x, t)
then furnishes curvilinear coordinates on the 2D domain Ω , on which the plane strain
shell problem is defined. We sometimes use the notation (x1 , x2 ) to replace (x, t) for
convenience. For brevity, the derivative ∂x will be denoted by ∂. The boundary of Ω is
composed of the upper and lower sides Γ± = Φ((0, L)×{± }) where the shell is subjected
to surface forces, and the lateral sides Γ0 = Φ({0} × [− , ]) and ΓL = Φ({L} × [− , ])
where the shell is clamped.
27
The curvature of S at the point φ(x) is defined by b(x) = a2 · ∂a1 . We denote
the maximum absolute value of the curvature by B = maxx∈[0,L] |b(x)|.
With the curvilinear coordinates defined on Ω , the covariant basis vectors at
(x, t) in Ω are g 1 = (1 − bt)a1 ,
g 2 = a2 . The covariant metric tensor gαβ is given
by g11 = (1 − bt)2 , g22 = 1, g12 = g21 = 0, and the contravariant metric tensor g αβ is
given by g 11 = 1/(1 − bt)2 , g 22 = 1, g 12 = g 21 = 0.
We denote the determinant of the covariant metric tensor by g = det(gαβ ). Then
the Jacobian of the transformation Φ is
Z
Ω
f ◦ Φ−1 =
√
g = 1 − bt. Therefore,
Z LZ f (x, t)(1 − bt)dtdx
0
(2.2.11)
−
holds for all f : ω → R. Often, we will simply write
Z
Ω
Z
f instead of
Ω
f ◦ Φ−1 .
Γ+
S
Γ0
Γ−
Ω
ΓL
Φ
φ
t
L
0
−
ω
Fig. 2.1. A cylindrical shell and its cross section
x
28
The Christoffel symbols of this metric are
Γ∗1
11 =
−∂bt
,
1 − bt
Γ∗1
12 =
Γ∗2
11 = b(1 − bt),
−b
,
1 − bt
Γ∗2
12 = 0,
Γ∗1
22 = 0,
Γ∗2
22 = 0.
The geometric equation becomes
χ11 (v ) = ∂v1 +
∼
∂bt
v − b(1 − bt)v2 ,
1 − bt 1
χ22 (v ) = ∂t v2 ,
∼
(2.2.12)
1
b
χ12 (v ) = χ21 (v ) = (∂t v1 + ∂v2 ) +
v .
∼
∼
2
1 − bt 1
The row divergence of a tensor field σ αβ , by (2.2.3), has the expression
σ 1β kβ = ∂σ 11 + ∂t σ 12 − 2
∂bt 11
b
σ −3
σ 12 ,
1 − bt
1 − bt
σ 2β kβ = ∂σ 12 + ∂t σ 22 + b(1 − bt)σ 11 −
(2.2.13)
∂bt 12
b
σ −
σ 22 .
1 − bt
1 − bt
Let the surface force densities on Γ± be p ± = pα
± g α , the body force density be
∼
q = q α g α . The equilibrium equation is
∼
σ αβ kβ + q α = 0.
(2.2.14)
The traction boundary conditions on Γ± expressed in terms of the contravariant components of a stress field σ read
∼
σ 12 ( · , ) = p1+ ,
σ 12 ( · , − ) = −p1− ,
σ 22 ( · , ) = p2+ ,
σ 22 ( · , − ) = −p2− . (2.2.15)
29
According to the definition, a stress field σ is statically admissible if both the
∼
equations (2.2.14) and (2.2.15) are satisfied by its contravariant components.
The clamping boundary condition imposed on an admissible displacement field
v (x, t) is simply
∼
v1 (0, · ) = v1 (L, · ) = v2 (0, · ) = v2 (L, · ) = 0.
2.2.4
(2.2.16)
Rescaled stress and displacement components
To simplify the calculation, we introduce the rescaled components σ̃ αβ for a stress
tensor σ αβ by
σ̃ 11 = (1 − bt)2 σ 11 ,
σ̃ 12 = (1 − bt)σ 12 ,
σ̃ 22 = (1 − bt)σ 22 .
(2.2.17)
Then
σ 1β kβ =
1
[∂ σ̃ 11 + (1 − bt)∂t σ̃ 12 − 2bσ̃ 12 ],
(1 − bt)2
σ 2β kβ =
(2.2.18)
1
[∂ σ̃ 12 + ∂t σ̃ 22 + bσ̃ 11 ],
1 − bt
which is noticeably simpler than (2.2.13).
In these curvilinear coordinates, and in terms of the rescaled stress components,
the constitutive equation
χαβ = Aαβλγ σ λγ
30
takes the form
χ11 =
2µ + λ
λ
(1 − bt)2 σ̃ 11 −
(1 − bt)σ̃ 22 ,
4µ(µ + λ)
4µ(µ + λ)
1
(1 − bt)σ̃ 12 ,
χ12 = χ21 =
2µ
χ22 =
(2.2.19)
2µ + λ
1
λ
σ̃ 22 −
σ̃ 11 .
4µ(µ + λ) 1 − bt
4µ(µ + λ)
For consistency with the rescaled stress components, we introduce the rescaled
components q̃ α for the body force density and rescaled components p̃α for the surface
force density.
For the body force density, we define the rescaled components by
q = q α g α = q̃ α
∼
1
aα .
1 − bt
(2.2.20)
In components, we have q̃ 1 = (1 − bt)2 q 1 and q̃ 2 = (1 − bt)q 2 . The rescaled components
account the area change in the transverse direction of the cross section and more explicitly
reflect the variation of the body force density in that direction. We define the components
of the transverse average and moment of the body force density by
qaα =
Z 1
q · aα dt,
2 − ∼
α =
qm
Z 3
t q · aα dt.
3
2 − ∼
(2.2.21)
In the following, we assume the body force density changes linearly in t, or equivalently,
α )a . Under this assumption, the rescaled components are quadratic
q = (qaα + tqm
α
∼
31
α − bq α , and
polynomials in t, and we have q̃ α = q0α + tq1α + t2 q2α , with q0α = qaα , q1α = qm
a
α.
q2α = −bqm
The ensuing calculations can be carried through if q̃ α are arbitrary quadratic
polynomials in t. Without this restriction, we cannot apply the two energies principle
directly. For a general body force density, the convergence of the model can be proved
under some restriction on the transverse variation of the body force density. This issue
will be addressed in the general shell theory.
For the surface force density p ± , we introduce the rescaled components p̃α
± by
∼
1
α
p = pα
+ g α = p̃+ 1 − b g α ,
∼+
1
α
p = pα
− g α = p̃− 1 + b g α .
∼−
(2.2.22)
The rescaled components account the length differences of the upper and lower curves
of the shell cross section from middle curve. In terms of the rescaled surface force
components, we define
p̃1 − p̃1−
,
p1o = +
2
p̃2 − p̃2−
p2o = +
,
2
p̃1 + p̃1−
p1e = +
,
2
p̃2 + p̃2−
p2e = +
,
2
(2.2.23)
which are the odd and weighted even parts of the upper and lower surface forces.
In terms of the rescaled stress components σ̃ αβ and the rescaled applied force
components, the equilibrium equation (2.2.14) and the surface force condition (2.2.15)
can be written as
∂ σ̃ 11 + (1 − bt)∂t σ̃ 12 − 2bσ̃ 12 + q̃ 1 = 0,
(2.2.24)
∂ σ̃ 12 + ∂t σ̃ 22 + bσ̃ 11 + q̃ 2 = 0
32
and
σ̃ 12 ( · , ± ) = p1o ± p1e ,
σ̃ 22 ( · , ± ) = p2o ± p2e .
(2.2.25)
We introduce the rescaled displacement components ṽα for the displacement vector v by expressing it as the combination of basis vectors on the middle curve, i.e.,
∼
v = vα g α = ṽα aα , or equivalently, v1 = (1 − bt)ṽ1 ,
∼
v2 = ṽ2 . In terms of the rescaled
components ṽα , by using (2.2.12), the geometric equation becomes
χ11 (v ) = (1 − bt)(∂ṽ1 − bṽ2 ),
∼
χ22 (v ) = ∂t ṽ2 ,
∼
(2.2.26)
1
χ12 (v ) = χ21 (v ) = [bṽ1 + ∂ṽ2 + (1 − bt)∂t ṽ1 ].
∼
∼
2
And the clamping boundary condition is
ṽα (0, · ) = ṽα (L, · ) = 0.
(2.2.27)
In summary, in terms of the rescaled components, the elasticity problem seeks displacement components ṽα and stress components σ̃ αβ satisfying the constitutive equation (2.2.19), the equilibrium equation (2.2.24), the geometric equation (2.2.26) and the
boundary conditions (2.2.25) and (2.2.27).
2.3
The shell model
Our shell model is a 1D variational problem defined on the space H = [H01 (0, L)]3 .
The solution of the model is composed of three single variable functions that approximately describe the shell displacement arising in response to the applied forces and
33
boundary conditions. For any (θ, u, w) ∈ H, we define
γ(u, w) = ∂u − bw, ρ(θ, u, w) = ∂θ + b(∂u − bw), τ (θ, u, w) = θ + ∂w + bu, (2.3.1)
which give the membrane strain, flexural strain and shear strain engendered by the
displacement functions (θ, u, w).
The model reads: Find (θ , u , w ) ∈ H, such that
Z L
1 2
?
(2µ + λ )
ρ(θ , u , w )ρ(φ, y, z)dx
3
0
Z L
Z L
5
?
+ (2µ + λ )
γ(u , w )γ(y, z)dx + µ
τ (θ , u , w )τ (φ, y, z)dx
6 0
0
= hf 0 + 2 f 1 , (φ, y, z)i ∀(φ, y, z) ∈ H, (2.3.2)
in which
λ? =
2µλ
,
2µ + λ
and the resultant loading functionals are given by
Z
Z L
5 L 1
λ
hf 0 , (φ, y, z)i =
p τ (φ, y, z)dx −
p2 γ(y, z)
6 0 o
2µ + λ 0 o
Z L
+
[(p1e + qa1 − 2bp1o )y + (p2e + qa2 + ∂p1o )z]dx (2.3.3)
0
and
Z
1 L
1 )φ + bq 1 y + bq 2 z]dx
[(bqa1 + 3bp1e − qm
hf 1 , (φ, y, z)i = −
m
m
3 0
34
Z L
Z
1 L 1
λ
2
2
(pe + bpo )ρ(φ, y, z)dx −
bpe τ (φ, y, z)dx. (2.3.4)
−
3(2µ + λ) 0
6 0
The bilinear form in the left hand side of the variational formulation of the model
(2.3.2) is uniformly elliptic in the space H = [H01 (0, L)]3 . This conclusion follows from
the following theorem.
Theorem 2.3.1. The equivalency
kρ(θ, u, w)kL (0,L) + kγ(u, w)kL (0,L) + kτ (θ, u, w)kL (0,L) ' k(θ, u, w)kH
2
2
2
(2.3.5)
holds for all (θ, u, w) ∈ H = [H01 (0, L)]3 . Here ρ, γ and τ are the strain operators defined
in (2.3.1).
To prove this result, we need Peetre’s lemma.
Lemma 2.3.2. Let X, Y1 , Y2 be Hilbert spaces, and let A1 : X → Y1 and A2 : X → Y2
be bounded linear operators with A1 injective and A2 compact. If there exists a constant
c > 0 such that
kxkX ≤ c(kA1 xkY1 + kA2 xkY2 )
∀x ∈ X,
then there exists a constant c0 > 0 such that
kxkX ≤ c0 kA1 xkY1
∀x ∈ X.
For a proof of this lemma, see [28]. We give the proof of the theorem.
35
Proof of Theorem 2.3.1. The upper bound of the left hand side is obvious. For
the lower bound, we first see that
kρ(θ, u, w)kL (0,L) + (1 + B)kγ(u, w)kL (0,L) + kτ (θ, u, w)kL (0,L)
2
2
2
≥ k∂θkL (0,L) + k∂u − bwkL (0,L) + k∂w + θ + bukL (0,L) .
2
2
2
We consider the operators A1 and A2 from H to [L2 (0, L)]3 defined by,
A1 (θ, u, w) = (∂θ, ∂u − bw, ∂w + θ + bu),
A2 (θ, u, w) = (0, bw, θ + bu), ∀ (θ, u, w) ∈ H.
The operator A1 is injective, since if (θ, u, w) ∈ ker A1 , then θ = 0, ∂u − bw = 0 and
∂w + bu = 0, so u∂u + w∂w = 0, therefore, u2 + w2 = constant. Since u and w vanish on
the end points of the interval, we must have u = w = 0. The operator A2 is obviously
compact. The statement follows from Lemma 2.3.2.
Theorem 2.3.1 shows that if the resultant loading functional f 0 + 2 f 1 is in the dual
space of H, the model problem is uniquely solvable.
Remark 2.3.1. The requirement f 0 + 2 f 1 ∈ H ∗ can be met, if, say, the the applied
force functions are square integrable. To prove the convergence, we will need to assume
the tangential surface forces p̃1± ∈ H 1 (0, L). To prove the best possible convergence rate,
we will further need to assume the normal surface forces p̃2± ∈ H 1 (0, L). Henceforth, we
will assume that
1
α α
p̃α
± ∈ H (0, L), qa , qm ∈ L2 (0, L).
(2.3.6)
36
This model is slightly different from that of Naghdi’s in the following aspects:
1. There is a shear correction factor 5/6. The best value for this factor is an
unresolved issue in shell theories. For the special case of plate, the value 5/6 is usually
accepted as the best. We will see that in the flexural case, the problem is not sensitive
to this value. In the case of membrane–shear, if this factor is changed, there must be a
corresponding change in the resultant loading functional, otherwise a poor choice of the
factor may lead to divergence of the model.
2. The expression for flexural strain is ∂θ + b(∂u − bw) while in the classical
Naghdi model it is ∂θ − b(∂u − bw). This change of the flexural strain operator rooted in
our derivation of the model, in which, the dimensionally reduced constitutive equation
was derived by roughly minimizing constitutive residual. Our choice leads to a smaller
constitutive residual. See Remark 2.4.1. Aother evidence favoring this change is provided
by modeling a semi-circular cylindrical shell, in which this change is simply a consequence
of more accurate integrations in the transverse direction in the process of classical Naghdi
model derivation.
3. The resultant loading functional contains more information than is normally
retained in the Naghdi model. The model convergence and convergence rate in the
relative energy norm can be proved if only f 0 is kept in the loading functional. See
Section 7.1.
37
2.4
Reconstruction of the stress and displacement fields
From the model solution (θ , u , w ) ∈ [H01 (0, L)]3 , we can rebuild a statically
admissible stress field by explicitly giving its contravariant components, and a kinematically admissible displacement field by giving its covariant components. We will prove the
convergence of both the reconstructed stress field and displacement field to the actual
fields determined from the 2D elasticity equations in the shell. The convergence will
be proved by using the two energies principle. To this end, we need to compute the
constitutive residual. We will see that the residual is formally small. Knowledge of the
behavior of the model solution will be necessary for a rigorous proof of the convergence.
2.4.1
Reconstruction of the statically admissible stress field
For brevity, we denote the flexural, membrane, and shear strains engendered by
the model solution by
ρ = ρ(θ , u , w ),
γ = γ(u , w ),
τ = τ (θ , u , w ).
We define three single variable functions σ111 , σ011 , and σ012 by
σ111 = (2µ + λ? )ρ +
λ
(p2 + bp2o ),
2µ + λ e
1
λ
σ011 = b 2 σ111 + (2µ + λ? )γ +
p2 ,
3
2µ + λ o
5
5
1
σ012 = µτ − p1o + b2 p1e ,
4
4
4
(2.4.1)
38
which furnish the principal part of the statically admissible stress field. It is straightforward to verify that these functions satisfy the following equations:
1 2 11 2 12
1
1 ),
∂σ1 − σ0 = 2 bp1e + 2 (bqa1 − qm
3
3
3
2
1
1,
∂σ011 − bσ012 = 2bp1o − p1e − qa1 + 2 bqm
3
3
(2.4.2)
1
2
2.
bσ011 + ∂σ012 = −p2e − ∂p1o − qa2 + 2 bqm
3
3
Actually, by substituting (2.4.1) into (2.4.2), we will get a system of three second order
ordinary differential equations, which is just the differential form of the variational model
equation (2.3.2). Obviously, the three principal stress functions are in L2 (0, L). Furthermore, the equations in (2.4.2) clearly show that these three functions are in H 1 (0, L).
To complete the construction of a statically admissible stress field, we also need
three supplementary functions σ211 , σ022 , and σ122 . They are defined by
1,
∂σ211 = −4bσ012 + 2 bqm
1
2 − bq 2 ),
σ022 = 2 (bσ111 + ∂p1e + qm
a
2
(2.4.3)
1 2
2 ).
σ122 = ( bσ211 + bσ011 + p2e + ∂p1o + qa2 − 2 bqm
2 3
Note that the first equation in (2.4.3) only gives ∂σ211 , so σ211 is determined up to an
Z L
arbitrary additive constant. We fix a particular solution by requiring
σ211 = 0. Then
0
1 k ).
kσ211 kH 1 (0,L) . B(kσ012 k0 + 2 kqm
0
(2.4.4)
39
With these six functions determined, the rescaled stress components σ̃ αβ then
are explicitly defined by
σ̃ 11 = σ011 + tσ111 + r(t)σ211 ,
σ̃ 12 = σ̃ 21 = p1o + tp1e + q(t)σ012 ,
(2.4.5)
σ̃ 22 = p2o + tp2e + q(t)σ022 + s(t)σ122 ,
where
1
t2
r(t) = 2 − ,
3
t2
q(t) = 1 − 2 ,
t
t2
s(t) = (1 − 2 ).
(2.4.6)
Note that r is an even function of t and has zero integral over the interval [− , ], and
q(± ) = s(± ) = 0. Following classical terminology, we will call σ011 the resultant
membrane stress, σ111 the first membrane stress moment, and σ211 the second membrane
stress moment. The function σ012 is responsible for the quadratic distribution of the
rescaled shear stress in the transverse direction and will be shown to be a higher order
term. The two functions σ022 and σ122 enrich the variation of the normal stress in the
transverse direction.
With this choice of the rescaled stress components, the surface traction condition
(2.2.25) is precisely satisfied. Combining the six equations in (2.4.2) and (2.4.3) and the
definition (2.4.5), we can verify that the equilibrium equation (2.2.24) is precisely satisfied. Therefore, by the relation between the rescaled components and the contravariant
components (2.2.17), we get the contravariant components σ αβ of a statically admissible
40
stress field σ .
∼
σ 11 =
1
[σ 11 + tσ111 + r(t)σ211 ],
(1 − bt)2 0
σ 12 = σ 21 =
σ 22 =
2.4.2
1
[p1 + tp1e + q(t)σ012 ],
1 − bt o
(2.4.7)
1
[p2 + tp2e + q(t)σ022 + s(t)σ122 ].
1 − bt o
Reconstruction of the kinematically admissible displacement field
The rescaled components of the displacement field are defined by
ṽ1 = u + tθ ,
ṽ2 = w + tw1 + t2 w2 .
(2.4.8)
Here, w1 ∈ H01 (0, L) and w2 ∈ H01 (0, L) are two correction functions defined as solutions
of the following equations.
λ
1
[p2o −
σ 11 ], v)L (0,L)
2 (∂w1 , ∂v)L (0,L) + (w1 , v)L (0,L) = (
?
2
2
2
2µ + λ
2µ + λ 0
(2.4.9)
∀ v ∈ H01 (0, L)
and
1
λ
2 (∂w2 , ∂v)L (0,L) + (w2 , v)L (0,L) = (
[p2e −
σ 11 ], v)L (0,L)
?
2
2
2
2(2µ + λ )
2µ + λ 1
∀ v ∈ H01 (0, L).
(2.4.10)
The clamping boundary condition (2.2.27) is obviously satisfied. Note that this correction does not affect the middle curve displacement. So the basic pattern of the shell
41
deformation is already well captured by the model solution. The covariant components
of the kinematically admissible displacement field v are
∼
v1 = (1 − bt)(u + tθ ),
v2 = w + tw1 + t2 w2 .
(2.4.11)
These components are in H 1 (ω ), and satisfy the requirement of the two energies principle.
2.4.3
Constitutive residual
We denote the residual of the constitutive equation by %αβ = Aαβλγ σ λγ −
χαβ (v ), in which σ αβ and vα are the components of the admissible stress and dis∼
placement fields constructed from the model solution in the previous subsections.
By the formulae (2.2.26), we have
χ11 (v ) = (1 − bt)(∂u + t∂θ − bw − btw1 − bt2 w2 )
∼
= γ + tρ − 2btγ − b(1 − bt)(tw1 + t2 w2 ) − bt2 ∂θ ,
1
χ12 (v ) = χ21 (v ) = (θ + ∂w + bu + t∂w1 + t2 ∂w2 )
∼
∼
2
1 1
= τ + (t∂w1 + t2 ∂w2 ),
2
2
χ22 (v ) = w1 + 2tw2 .
∼
(2.4.12)
42
By the formulae (2.2.19), the definitions (2.4.1) and (2.4.5), and the identity (2µ +
λ)/[4µ(µ + λ)] = 1/(2µ + λ? ), we have
A11λγ σ λγ = γ + tρ − 2btγ +
1
1
{b2 t2 [σ011 + tσ111 + r(t)σ211 ] + [ b 2 (1 − 2bt) − 2bt2 ]σ111 }
2µ + λ?
3
−
λ
{(1 − bt)[q(t)σ022 + s(t)σ122 ] − bt2 p2e }
4µ(µ + λ)
+
1
(1 − 2bt)r(t)σ211 ,
2µ + λ?
A12λγ σ λγ =
1
(1 − bt)[p1o + tp1e + q(t)σ012 ],
2µ
A22λγ σ λγ =
1
λ
1
λ
(p2 −
σ 11 ) + t
(p2 −
σ 11 )
2µ + λ? o 2µ + λ 0
2µ + λ? e 2µ + λ 1
(2.4.13)
+
1
bt
{q(t)σ022 + s(t)σ122 +
[p2 + tp2e + q(t)σ022 + s(t)σ122 ]}
?
2µ + λ
1 − bt o
−
λ
r(t)σ211 .
4µ(µ + λ)
Subtracting (2.4.12) from (2.4.13), we obtain the following expressions for the constitutive residual:
%11 =
1
1
{b2 t2 [σ011 + tσ111 + r(t)σ211 ] + [ b 2 (1 − 2bt) − 2bt2 ]σ111 }
?
2µ + λ
3
−
λ
{(1 − bt)[q(t)σ022 + s(t)σ122 ] − bt2 p2e }
4µ(µ + λ)
+ b(1 − bt)(tw1 + t2 w2 ) + bt2 ∂θ
+
%12 =
1
(1 − 2bt)r(t)σ211 ,
2µ + λ?
1 5
1
[ q(t) − 1](µτ − p1o ) − (t∂w1 + t2 ∂w2 )
2µ 4
2
(2.4.14)
43
+
%22 = [
1
1
1
[t + q(t)b 2 ]p1e −
bt[p1o + tp1e + q(t)σ012 ]
2µ
4
2µ
(2.4.15)
λ
λ
1
1
(p2o −
(p2e −
σ011 ) − w1 ] + t[
σ 11 ) − 2w2 ]
?
?
2µ + λ
2µ + λ
2µ + λ
2µ + λ 1
+
1
bt
{q(t)σ022 + s(t)σ122 +
[p2 + tp2e + q(t)σ022 + s(t)σ122 ]}
?
2µ + λ
1 − bt o
−
λ
r(t)σ211 .
4µ(µ + λ)
(2.4.16)
Remark 2.4.1. If we had not made the sign change in the flexural strain ρ(θ, u, w)
discussed earlier, there would be an additional term 2btγ(u , w ) in the residual %11 .
Our variant does make the residual smaller, at least formally.
Formally, most of the terms in the above residual expressions contain a factor
of the form , t or smaller (recall that σ022 and σ122 have a small factor in their own
expressions (2.4.3)). In the expression of %11 , the only term not formally small is the
last one, whose magnitude is determined by that of σ211 . The big term in the expression
of %12 is in the first one, which is determined by
µτ − p1o .
(2.4.17)
This term is also the dominant part in the expression of σ012 , see (2.4.1). We will prove
that µτ − p1o is indeed small. Therefore, σ012 is small, and by (2.4.4), so is σ211 .
The definitions (2.4.9) and (2.4.10) of the correction functions w1 and w2 were
made to minimize the first two terms in the expression of %22 , at the same time, they
44
minimize the two terms t∂w1 and t2 ∂w2 in the expression of %12 . Therefore, we shall
be able to show that %22 is small as well.
2.5
Justification
The formal observations we made in the previous section do not furnish a rigorous
justification, since the applied forces and the model solution may depend on the the shell
thickness. To prove the convergence, we need to make some assumptions on the applied
loads, and get a good grasp of the behavior of the model solution when the shell thickness
tends to zero. Since we wish to bound the relative error, in addition to the upper bound
that can be determined from the constitutive residual, we need to have a lower bound
on the model solution.
2.5.1
Assumption on the applied forces
Henceforth, we assume that all the applied force functions explicitly involved in
the resultant loading functional of the model are independent of , i.e., the single variable
functions
α α
α
pα
o , pe , qa , and qm are independent of .
(2.5.1)
This assumption is different from the usual assumption adopted in asymptotic theories,
α
according to which, the functions −1 pα
o , rather than po themselves, should have been
α
α
assumed to be independent of . Our assumption on pα
e , qa and qm is the same as the
usual one. This different assumption will reveal the potential advantages of the Naghditype model over the Koiter-type model. The convergence theorem can also be proved
45
under the usual assumption on the applied forces, but it can be proved that the difference
between the two types of models then is negligible.
2.5.2
An abstract theory
Under the assumption (2.5.1) on the applied forces, the model (2.3.2) is an -
dependent variational problem fitting into the abstract problem that we shall discuss
in Chapter 3, cf., (3.2.2). The following convergence bounds (2.5.4) and (2.5.7) easily
follow from Theorem 3.3.1.
Let U, V , and H be Hilbert spaces, A : H → U a bounded linear operator, and
B : H → V a bounded linear continuous surjection. We assume that
kAukU + kBukV ' kukH ∀ u ∈ H.
(2.5.2)
For any f0 , f1 ∈ H ∗ and f0 6= 0, we consider the variational problem
2 (Au, Av)U + (Bu, Bv)V = hf0 + 2 f1 , vi,
(2.5.3)
u ∈ H, ∀v ∈ H.
It is obvious that under the equivalency assumption (2.5.2), this variational problem has
a unique solution u ∈ H that is dependent on . When → 0, the behavior of the
solution u is drastically different depending on whether f0 |ker B is nonzero or not. As
we shall see, in the former case, the solution u blows up at the rate of O(−2 ), while in
the latter case u tends to a finite limit.
46
For the first case, to get more accurate description of the behavior of the solution, we rescale the problem by assuming f0 = 2 F0 and f1 = 2 F1 with F0 , F1 ∈ H
independent of . Under this assumption, we have the convergence estimate
kAu − Au0 kU + −1 kBu kV . kF0 kH ∗ + 2 kF1 kH ∗ ,
(2.5.4)
in which u0 ∈ ker B is independent of and is the solution of the limit problem
(Au0 , Av)U = hF0 , vi ∀ v ∈ ker B.
(2.5.5)
Since F0 |ker B 6= 0, we must have Au0 6= 0.
For the second case, since f0 ∈ (ker B)a (the annihilator of ker B) and B is
surjective, there exists a unique ζ 0 ∈ V , such that
hf0 , vi = (ζ 0 , Bv)V ∀ v ∈ H.
(2.5.6)
In this case, there exists a unique u0 ∈ H such that Bu0 = ζ 0 , and we have the
convergence estimate
kAu − Au0 kU + −1 kBu − ζ 0 kV . (kf0 kH ∗ + kf1 kH ∗ ).
(2.5.7)
It can be shown that the limit u0 can be determined as u0 = u00 + u01 . Here
(Au00 , Av)U + (Bu00 , Bv)V = 0 ∀v ∈ ker B, i.e., u00 is in the orthogonal complement of
ker B in H with respect to the inner product (A · , A · )U + (B · , B · )V that, due to the
47
equivalency assumption (2.5.2), is equivalent to the original inner product of H. And
u01 ∈ ker B is the solution of the limit problem corresponding to f1 ,
(Au01 , Av)U = hf1 , vi ∀ v ∈ ker B.
(2.5.8)
Since f0 6= 0, we have ζ 0 =
6 0.
2.5.3
Asymptotic behavior of the model solution
To fit the model problem (2.3.2) in the abstract framework (2.5.3), we introduce
the following Hilbert spaces,
H = [H01 (0, L)]3 ,
U = L2 (0, L),
V = [L2 (0, L)]2 .
The inner product in H is the usual one. The inner products in U and V will be changed
slightly and equivalently. For ρ1 , ρ2 ∈ U , we define
1
(ρ1 , ρ2 )U = (2µ + λ? )(ρ1 , ρ2 )L (0,L)
2
3
and for [γ1 , τ1 ], [γ2 , τ2 ] ∈ V , we define
5
([γ1 , τ1 ], [γ2 , τ2 ])V = (2µ + λ? )(γ1 , γ2 )L (0,L) + µ(τ1 , τ2 )L (0,L) .
2
2
6
We define the operators by
A(θ, u, w) = ρ(θ, u, w) ∀ (θ, u, w) ∈ H,
48
which is just the flexural strain operator, and
B(θ, u, w) = [γ(u, w), τ (θ, u, w)] ∀ (θ, u, w) ∈ H,
which combines the membrane and shear strains engendered by the displacement functions.
The equivalence (2.3.5) that was established in Theorem 2.3.1 guaranteed the
condition (2.5.2). To use the abstract results, we also need to show that the operator B
is surjective. To this end, it is convenient to consider the dual operator B ∗ of B. It is
easy to see that
B ∗ : [L2 (0, L]2 −→ [H −1 (0, L)]3 ,
B ∗ (ζ, η) = (η, bη − ∂ζ, −∂η − bζ) ∀ (ζ, η) ∈ [L2 (0, L]2 .
We have
Lemma 2.5.1. If the curvature b of the middle curve S of the cross section of the cylindrical shell is not identically equal to zero, then the dual operator B ∗ is injective and has
closed range.
Proof. If (ζ, η) ∈ ker B ∗ , then
kηk−1 = 0,
kbη − ∂ζk−1 = 0,
and k∂η + bζk−1 = 0,
49
so we have
η = 0,
and k∂ζk−1 = 0,
kbζk−1 = 0.
Since the curvature b is not identically equal to zero, we must have ζ = 0.
By viewing B ∗ as the operator A1 in Lemma 2.3.2, and considering the compact
operator
A2 : [L2 (0, L)]2 −→ [H −1 (0, L)]3
defined by A2 (η, ζ) = (0, bη, bζ), the desired result will follow from lemma 2.3.2.
The statement that the operator B is surjective then follows from the closed range
theorem.
Remark 2.5.1. If the curvature b is identically equal to zero, the operator B is still
surjective, but the range will be [L2 (0, L)/R] × L2 (0, L). All the results of this section
still apply.
In accordance with the abstract theory, when the shell thickness tends to zero,
the behavior of the model solution (θ , u , w ) can be dramatically different for whether
f 0 |ker B 6= 0
(2.5.9)
f 0 |ker B = 0.
(2.5.10)
or
50
We assume f 0 6= 0, otherwise, the model is reduced down to a problem loaded by 2 f 1 ,
and all the analysis can be likewisely carried out and the convergence theorem in the
relative energy norm can also be proved.
a. Undeformed
c. Membrane deformation
b. Flexural deformation
d. Shear deformation
Fig. 2.2. Deformations of a cylindrical shell
Since the geometry of the middle surface of a cylindrical shell and the two sides
clamping boundary condition together do not inhibit pure flexural deformation (ker B 6=
0), a plane strain cylindrical shell problem can be classified as a flexural shell. However
the behavior of the shell is very different depending on whether or not the applied forces
make the pure flexural deformation happen. Similar situations for the second case arise
in stretching a plate, or twisting a plate by tangential surface forces that are equal in
magnitude but opposite in direction on the upper and lower surfaces. If the applied
forces do bring about the non-inhibited asymptotically pure flexural deformation, see
Figure 2.2 (b), the flexural energy will dominate membrane and shear strain energies.
51
If the applied force does not make the pure flexural deformation happen, as shown by
Figure 2.2 (c) (d), the membrane and shear strain energies together will dominate the
flexural energy. Since their magnitudes might be the same, there is no way for us to
distinguish the membrane and shear energies. For this reason, and for consistency with
terminologies in general shell theory, we call the first case the case of flexural shells, and
the second one the membrane–shear shells.
For a flexural shell, the solution blows up at the rate of O(−2 ). To get an
accurate grasp of the model solution behavior, we need to scale the loading functional
as we did for the abstract problem. This scaling is equivalently imposed on the applied
force functions by assuming
2 α
pα
o = Po ,
2 α
pα
e = Pe ,
qaα = 2 Qα
a,
α = 2 Qα ,
qm
m
(2.5.11)
α
with Poα , Peα , Qα
a , Qm single variable functions independent of . The resultant loading
functionals are accordingly scaled as f 0 = 2 F 0 ,
f 1 = 2 F 1 , with F 0 and F 1
independent of . The expressions for F 0 and F 1 are the same as (2.3.3) and (2.3.4),
had the lower case letters been replaced by capital letters. By the estimate (2.5.4) we
have
kρ − ρ0 kL (0,L) + −1 kγ kL (0,L) + −1 kτ kL (0,L) . ,
2
2
2
(2.5.12)
52
in which ρ0 = ρ(θ0 , u0 , w0 ), and (θ0 , u0 , w0 ) ∈ ker B is the solution of the limit problem
Z L
1
?
ρ(θ0 , u0 , w0 )ρ(φ, y, z)dx = hF 0 , (φ, y, z)i
(2µ + λ )
3
0
∀ (φ, y, z) ∈ ker B,
(2.5.13)
(θ0 , u0 , w0 ) ∈ ker B.
This limit problem is nothing else but the limit flexural shell model. Since F 0 |ker B 6= 0,
we have ρ0 =
6 0.
For a membrane–shear shell, when → 0, the model solution (θ , u , w ) converges
to a finite limit. In this case, the resultant loading functional can be reformulated as
hf 0 , (φ, y, z)i = (ζ 0 , B(φ, y, z))V ∀ (φ, y, z) ∈ H.
Note that if the curvature b is not identically equal to zero, the strain operator γ(u, w) =
∂u − bw is surjective from [H01 (0, L)]2 to L2 (0, L). Recalling the expression (2.3.3):
Z
Z L
λ
5 L 1
p τ (φ, y, z)dx −
p2 γ(y, z)
hf 0 , (φ, y, z)i =
6 0 o
2µ + λ 0 o
Z L
+
[(p1e + qa1 − 2bp1o )y + (p2e + qa2 + ∂p1o )z]dx,
0
we see that the condition f 0 |ker B = 0 is equivalent to that the bounded linear functional
defined by the second term in the right hand side of this equation vanishes on the kernel of
the strain operator γ. Therefore, by the closed range theorem, the condition f 0 |ker B = 0
53
is equivalent to the unique existence of γ 0 ∈ L2 (0, L), such that
hf 0 , (φ, y, z)i = (2µ + λ? )
Z L
Z L
5
1 1
γ 0 γ(y, z)dx + µ
p τ (φ, y, z)dx ∀ (φ, y, z) ∈ H.
6 0 µ o
0
Recalling the definition of inner product in the space V , it is readily seen that the
element ζ 0 ∈ V in the abstract theory takes the form
1
ζ 0 = (γ 0 , p1o ).
µ
By the estimate (2.5.7), we get
1
kρ − ρ0 kL (0,1) + −1 kγ − γ 0 kL (0,L) + −1 kτ − p1o kL (0,L) . ,
2
2
2
µ
(2.5.14)
in which ρ0 = ρ(θ0 , u0 , w0 ) with (θ0 , u0 , w0 ) ∈ H be the limit of the solution (θ , u , w ).
1
Actually, we also have γ 0 = γ(θ0 , u0 , w0 ) and p1o = τ (θ0 , u0 , w0 ).
µ
By their definitions (2.4.9), (2.4.10), and Theorem 3.3.6 of Chapter 3, we have
the following estimates on the correction functions w1 and w2 .
1
λ
k∂w1 kL (0,L) + k
σ 11 ) − w1 kL (0,L)
(p2 −
2
2
2µ + λ? o 2µ + λ 0
. 1/2 [kp2o kH 1 (0,L) + kσ011 kH 1 (0,L) ] (2.5.15)
54
and
1
λ
k∂w2 kL (0,L) + k
(p2e −
σ 11 ) − 2w2 kL (0,L)
?
2
2
2µ + λ
2µ + λ 1
. 1/2 [kp2e kH 1 (0,L) + kσ111 kH 1 (0,L) ]. (2.5.16)
2.5.4
Convergence theorem
With all the above preparations, we are ready to prove the convergence theorem.
We denote the energy norm of a stress field σ and a strain field χ defined on the shell
∼
∼
cross section Ω by
Z
kσ kE = (
∼
Ω
Aαβλγ σ λγ σ αβ )1/2
Z
and kχ kE = (
C αβλγ χλγ χαβ )1/2 ,
∼
Ω
respectively. Since the elasticity tensor C αβλγ and the compliance tensor Aαβλγ are
uniformly positive definite and bounded, the energy norms are equivalent to the sums of
the L2 (ω ) norms of the tensor components.
Theorem 2.5.2. Assume that the surface force functions have the regularity p̃α
± ∈
α ∈ L (0, L). Let σ ∗ be the actual stress
H 1 (0, L) and the body force functions qaα , qm
2
∼
distribution over the loaded shell, and v ∗ the true displacement field arising in response
∼
to the applied forces and boundary conditions. Based on the model solution (θ , u , w ),
we define the statically admissible stress field σ by the formulae (2.4.7), and define the
∼
kinematically admissible displacement field v by the formulae (2.4.11). We have the
∼
55
estimate
kσ ∗ − σ kE + kχ (v ∗ ) − χ (v )kE ∼
∼
∼ ∼
kχ (v )kE ∼ ∼
∼ ∼
. 1/2 .
(2.5.17)
Proof. The proof is based on the two energies principle, the formulae for the constitutive
residual (2.4.14) – (2.4.16), the asymptotic behaviors (2.5.12) and (2.5.14) of the model
solution, and the estimates (2.5.15) and (2.5.16) on the correction functions. Since the
behaviors of the model solution are very different for flexural shells and membrane–shear
shells, we prove the theorem for the two cases separately. In the following, we will simply
denote the norm k · kL (ω ) by k · k.
2
Flexural shells
This is the case in which the solution blows up at the rate of O(−2 ).
To
ease the analysis, we scale the loading functions by assuming that (2.5.11) holds, with
α
Poα , Peα , Qα
a , Qm single variable functions independent of . Note that, since we are con-
sidering the relative error estimate, this scaling is not a real restriction on the applied
force functions. With this scaling, we have the estimate (2.5.12), from which, we get
kρ − ρ0 kL (0,L) . ,
2
kγ kL (0,L) . 2 ,
2
kτ kL (0,L) . 2 .
2
(2.5.18)
kθ kH 1 (0,L) + ku kH 1 (0,L) + kw kH 1 (0,L) ' 1 ' kρ0 kL (0,L) .
2
(2.5.19)
From the equivalence (2.3.5), we get
56
By the definition (2.4.1), we have
1
λ
σ011 = b 2 σ111 + (2µ + λ? )γ +
2 Po2 ,
3
2µ + λ
σ111 = (2µ + λ? )ρ +
λ
2 (Pe2 + bPo2 ),
2µ + λ
(2.5.20)
5
5
1
σ012 = µτ − 2 Po1 + b4 Pe1 ,
4
4
4
from which, we have the estimates
kσ011 kL (0,L) . 2 ,
2
kσ111 kL (0,L) ' 1,
2
kσ012 kL (0,L) . 2 .
2
(2.5.21)
By the estimate (2.4.4), we have
kσ211 kH 1 (0,L) . 2 .
(2.5.22)
From the first and last equations of (2.4.2), we see the estimates
kσ011 kH 1 (0,L) . 2 ,
kσ111 kH 1 (0,L) ' 1.
(2.5.23)
From the last two equations in (2.4.3), we see the estimates
kσ022 kL (0,L) . 2 ,
2
kσ122 kL (0,L) . 3 .
2
(2.5.24)
57
By the estimates on the correction functions (2.5.15) and (2.5.16), we have
1
λ
k∂w1 kL (0,L) + k
(2 Po2 −
σ 11 ) − w1 kL (0,L) . 5/2 ,
?
2
2
2µ + λ
2µ + λ 0
(2.5.25)
1
λ
k∂w2 kL (0,L) + k
(2 Pe2 −
σ 11 ) − 2w2 kL (0,L) . 1/2 .
?
2
2
2µ + λ
2µ + λ 1
From the equation (2.4.12), we see that in the expression of χ11 (v ), the term
∼
tρ(θ , u , w ) dominates in L2 (ω ), and by (2.5.18), we get the lower bound kχ11 (v )k2 &
∼
3 , and so
kχ (v )k2E & 3 .
(2.5.26)
∼ ∼
By the two energies principle, we have
Z
kσ ∗ − σ k2E + kχ (v ∗ ) − χ (v )k2E =
C αβλγ %λγ %αβ
∼
∼
∼ ∼
∼ ∼
Ω
(2.5.27)
. k%11 k2 + k%12 k2 + k%22 k2 .
In the expression (2.4.14) of %11 , we can see that the square integrals over ω of all
the terms are bounded by O(5 ), therefore, we have k%11 k2 . 5 . From the expression
(2.4.15) of %12 , we see that the square integrals of all the terms are bounded by O(4 ),
and so we have k%12 k2 . 4 . From the expression (2.4.16) of %22 , we see the bounds are
O(4 ), and so k%22 k2 . 4 .
Therefore, by (2.5.27), we get the upper bound
kσ ∗ − σ k2E + kχ (v ∗ ) − χ (v )k2E . 4
∼
∼
∼ ∼
∼ ∼
(2.5.28)
58
The conclusion of the theorem for the case of flexural shells then follows from the lower
bound (2.5.26) and this upper bound.
Membrane–shear shells
α α
α
In this case, under the assumption that pα
o , pe , qa , and qm are independent of ,
the model solution (θ , u , w ) converges to a finite limit in the space H when → 0, so
we have
kθ kH 1 (0,L) + ku kH 1 (0,L) + kw kH 1 (0,L) . 1.
From the estimate (2.5.14), we get
kρ kL (0,L) . 1,
2
kγ − γ 0 kL (0,L) . 2 ,
2
kµτ − p1o kL (0,L) . 2 .
2
(2.5.29)
Since ζ 0 6= 0, we know that γ 0 and p1o can not be zero simultaneously. From the equation
(2.4.1) we see
kσ011 kL (0,L) . 1,
2
kσ111 kL (0,L) . 1,
2
kσ012 kL (0,L) . 2 .
2
(2.5.30)
By the estimate (2.4.4), we have
kσ211 kH 1 (0,L) . 2 .
(2.5.31)
From the first and last two equations of (2.4.2), we see the estimates
kσ011 kH 1 (0,L) . 1,
kσ111 kH 1 (0,L) . 1.
(2.5.32)
59
From the last two equations in (2.4.3), we see the estimates
kσ022 kL (0,L) . 2 ,
2
kσ122 kL (0,L) . .
2
(2.5.33)
By the estimates on the correction functions (2.5.15) and (2.5.16), we have
1
λ
k∂w1 kL (0,L) + k
(p2 −
σ 11 ) − w1 kL (0,L) . 1/2 ,
2
2
2µ + λ? o 2µ + λ 0
(2.5.34)
1
λ
k∂w2 kL (0,L) + k
(p2e −
σ 11 ) − 2w2 kL (0,L) . 1/2 .
?
2
2
2µ + λ
2µ + λ 1
From the equation (2.4.12), we see that in the expression of χ11 (v ), the term γ(u , w )
∼
1
dominates, and in the expression of χ12 (v ), the term τ (θ , u , w ) dominates. Asymp∼
2
totically, we have the equivalency
1
kγ kL (0,L) + kτ kL (0,L) ' kγ 0 kL (0,L) + k p1o kL (0,L) ' 1.
2
2
2
2
µ
(2.5.35)
We get the lower bound kχ11 (v )k2 + kχ12 (v )k2 & , and so
∼
∼
kχ (v )k2E & .
∼ ∼
(2.5.36)
In the expression (2.4.14) of %11 , we can see that the square integrals over ω of all
the terms are bounded by O(3 ), therefore, we have k%11 k2 . 3 . From the expression
(2.4.15) of %12 , we see all the terms are bounded by O(2 ), and so we have k%12 k2 . 2 .
From the expression (2.4.16) of %22 , we see the bound is k%22 k2 . 2 .
60
By the two energies principle, we have
kσ ∗ − σ k2E + kχ (v ∗ ) − χ (v )k2E . k%11 k2 + k%12 k2 + k%22 k2 . 2 .
∼
∼
∼ ∼
∼ ∼
(2.5.37)
The conclusion of the theorem for the case of membrane–shear shells then follows from
the lower bound (2.5.36) and this upper bound.
2.6
Shear dominated shell examples
To emphasize the necessity of using the Naghdi-type model in some cases, we give
two examples for which the model equations are explicitly solvable. For these problems,
the Koiter model and limiting flexural model only give solutions that are identically
equal to zero, while our Naghdi-type model can very well capture the shear dominated
deformations.
2.6.1
A beam problem
p+
∼
p−
∼
a. Undeformed
b. Shear dominated deformation
Fig. 2.3. Shear dominated deformation of a beam
61
We consider a special plane strain cylindrical shell whose cross section is a thin
rectangle with thickness 2 and length L = 1. The curvature of the middle curve then
is b ≡ 0. The applied forces are: q = 0, p ± = ±a1 . The leading term of the resultant
∼
∼
loading functional (2.3.3) is given by
Z
5 1
hf 0 , (φ, y, z)i =
τ (φ, y, z)dx.
6 0
The condition f 0 |ker B = 0 is obviously satisfied. Therefore the limiting flexural shell
model only gives a zero solution. So does Koiter’s model.
The model (2.3.2), written in differential form, reduces to
5
5
1
− 2 (2µ + λ? )∂ 2 θ + µ(θ + ∂w ) = ,
3
6
6
−(2µ + λ? )∂ 2 u = 0,
5
− µ∂(θ + ∂w ) = 0,
6
(2.6.1)
(θ , u , w ) ∈ [H01 (0, 1)]3 ,
which is just the Timoshenko beam bending and stretching model [3]. The solution is
given by
θ = c x(1 − x),
u = 0,
1
1
w = − c (x − )x(1 − x),
3
2
µ 16 2 µ(µ + λ) −1
where c = [ +
] . We see the convergences:
6
5(2µ + λ)
1
lim θ = 6x(1 − x),
µ
→0
1
1
lim w = −2 (x − )x(1 − x),
µ
2
→0
lim (θ + ∂w ) =
→0
1
.
µ
62
This is basically the asymptotic pattern of the exact deformation of the elastic body.
Note that the last convergence shows that the transverse shear strain tends to a finite
limit, a violation of the Kirchhoff–Love hypothesis.
2.6.2
A circular cylindrical shell problem
In this subsection, we consider a plane strain circular cylindrical shell problem.
The shell occupies an infinitely long circular cylinder whose thickness is 2 . The middle
curve of the cross section Ω is the unit circle whose curvature is b = −1. The shell is
loaded by surface forces whose densities are p ± = ±(1 ∓ )2 a1 , and a body force whose
∼
contravariant components are given by q 1 =
12 2
r(t),
(1 + t)2
q 2 = 0.
p+
∼
p−
∼
a. Undeformed
b. Deformed
Fig. 2.4. Shear dominated deformation of a circular cylinder
It can be verified that both the net force and net torque resulting from the applied
forces are zero. Therefore the surface and body forces are compatible and the problem
63
α , the resultant
is well defined. By using the formulae (2.2.21) to compute qaα and qm
loading functional in the model can be computed. We have
Z
5 2π
2
hf 0 + f 1 , (φ, y, z)i =
τ (φ, y, z)dx
6 0
1
+ 2 [
Z 2π
2 0
Z 2π
Z 2π
ydx − 2
τ (φ, y, z)dx + 2
0
φdx] + 4 hr, (φ, y, z)i,
0
where r is a functional independent of . The higher order term O(4 r) is provably
negligible. With this higher order term cutoff, we will have f 0 |ker B = 0. The model
solution then is
u = 0,
w = 0,
θ =
1
9 2
−
,
µ 5µ
which gives a displacement field that is purely rotational. The covariant components
of the displacement field provided by this model are v1 = (1 + t)tθ ,
v2 = 0. The
covariant components of the strain tensor engendered by this displacement field are, by
the formulae (2.2.26),
χ11 = 0,
χ22 = 0,
1
9 2
1
χ12 = θ =
−
.
2
2µ 10µ
(2.6.2)
It can be easily checked that the stress field whose contravariant components are
given by
σ 11 = 0,
σ 22 = 0,
σ 12 =
1
[1 − 2t + 3t2 − 2 2 ]
1+t
(2.6.3)
64
is statically admissible. By using the formulae (2.2.19), we see that for the admissible
stress field defined by (2.6.3),
1
A11λγ σ λγ = A22λγ σ λγ = 0, A12λγ σ λγ =
(1 + t)[1 − 2t + 3t2 − 2 2 ].
2µ
Therefore, the the constitutive residual can be bounded by
%11 = 0,
%22 = 0,
|%12 | . .
From the two energies principle, we know that the pure rotational displacement given
by the model is very close to the exact displacement of this circular cylinder arising in
response to the applied forces. The error in the relative energy norm is O(). The shear
strain and stress absolutely dominate all the other strain and stress components. For
this problem, the Koiter model and the limiting flexural shell model only give a solution
that is identically equal to zero. This is a case for which the Naghdi-type model is
indispensable.
65
Chapter 3
Analysis of the parameter dependent
variational problems
3.1
Introduction
The plane strain cylindrical shell model (2.3.2) that we justified in the last chapter
can be put in the form of the abstract -dependent variational problem (3.2.2) below, and
Theorem 3.3.1 was essential to the justification. This abstract problem also applies to
the spherical shell model and general shell model that we are going to derive and justify.
It is the purpose of this chapter to establish all the a priori estimates that are necessary
for our analyses. The behavior of the solution of such -dependent can be drastically
different in different circumstances. We will classify the problem on the abstract level at
the end of this chapter. Results that will be used to analyze the relations between our
model and other existing shell theories will also be given.
3.2
The parameter dependent problem and its mixed formulation
For a Hilbert space X, we denote its dual by X ∗ , and for any f ∈ X ∗ , we use
iX f ∈ X to denote its Riesz representation. The isomorphism πX : X → X ∗ is defined
as the inverse of iX , and is equal to iX ∗ under the usual identification of X and X ∗∗ .
66
Let H, U , and V be Hilbert spaces, A and B be
linear continuous operators
from H to U and V , respectively. We assume
kAuk2U + kBuk2V ' kuk2H ∀ u ∈ H.
(3.2.1)
By properly defining spaces and operators, the shell models we derive can be
written in the form of the -dependent variational problem:
2 (Au, Av)U + (Bu, Bv)V = hf0 + 2 f1 , vi,
(3.2.2)
u ∈ H,
∀ v ∈ H,
where f0 , f1 ∈ H ∗ are two functionals independent of , and f0 6= 0. It turns out that,
in all the cases we are going to analyze, when → 0, the asymptotic behavior of solution
of this variational problem is mostly determined by the leading term f0 . For this reason,
we first analyze the abstract problem
2 (Au, Av)U + (Bu, Bv)V = hf, vi,
(3.2.3)
u ∈ H,
∀ v ∈ H,
with f ∈ H ∗ independent of . The behavior of the solution of (3.2.2) will be obtained
by a simple argument once the simpler problem (3.2.3) is fully understood.
The variational problem (3.2.3) represents the Timoshenko beam bending model
and Reissner–Mindlin plate bending model, with u standing for the transverse deflection
of the middle surface and rotation of normal fibers, and Au the bending strain and Bu
67
the transverse shear strain engendered by u. The Koiter shell model, which adopts the
Kirchhoff–Love assumption and so ignores the transverse shear deformation, takes this
form, with the variable u representing the middle surface displacement, Au the flexural
strain, and Bu the membrane strain. The Naghdi shell model, and the variant we derive,
can be put in this form if we let u be the middle surface displacement and normal fiber
rotation. The operator A defines the flexural strain and B combines the transverse
shear and membrane strains engendered by u. The spaces H is a multiple L2 -based first
order Sobolev space, and U and V are equivalent to L2 or products of L2 . Referring to
the physical background of the abstract problem, we will call 2 (Au, Au)U the flexural
energy, and (Bu, Bu)V the membrane–shear energy engendered by the displacement
function u.
Under the assumption (3.2.1), for any f ∈ H ∗ , the variational problem (3.2.3) has
a unique solution depending on . In what follows, whenever the dependence needs to
be emphasized, the solution will be denoted by u . We are concerned with the behavior
of the solution of such problems, especially when is small.
If we set F = −2 f , the following rough estimate is obvious
ku kH . kF kH ∗ .
(3.2.4)
We will derive more accurate estimates on the solution of (3.2.3) by introducing
a mixed formulation. In what follows, we will need some basic results.
68
First we recall that if X and Y are Hilbert spaces with X ⊂ Y , and if X is dense
in Y , then the restriction operator defines an injection of Y ∗ onto a dense subspace of
X ∗ (and we usually identify Y ∗ with that dense subspace).
We next recall the sum and intersection constructions for Hilbert spaces. If Hilbert
spaces X and Y are both continuously included in a larger Hilbert space, then the
intersection X ∩ Y and the sum X + Y are themselves Hilbert spaces with the norms
kzkX∩Y = (kzk2X + kzk2Y )1/2 and kzkX+Y =
inf (kxk2X + kyk2Y )1/2 ,
z=x+y
and we have
Lemma 3.2.1. If in addition, X ∩ Y is dense in both X and Y , then the dual spaces X ∗
and Y ∗ can be viewed as subspace of (X ∩ Y )∗ and we have
X ∗ + Y ∗ = (X ∩ Y )∗ .
The operator B : H → V may have closed range in some problems, as in the cases
of Timoshenko beam bending model, the plain strain cylindrical shell model of Chapter 2
and other 1D models. This operator may have a range that is not closed in V , as in the
Reissner–Mindlin plate, Koiter and Naghdi shell models, as well as numerous singular
perturbation problems.
Let W = B(H) ⊂ V be the range of B, whose norm is defined by, for any ζ ∈ W ,
kζkW = inf kukH .
ζ=Bu
(3.2.5)
69
With this norm, W is a Hilbert space isomorphic to H/ ker B. This space plays a crucial
role in the following analysis. For the Reissner–Mindlin bending model of a totally
clamped plate, this space is H̊ (rot). Without loss of generality we may assume that W
∼
is dense in V , otherwise, we can just replace V by the closure of W in it.
Associated with a Hilbert space X and any positive number ς, we define the
Hilbert spaces ςX. As set, ςX equals to X, but the norm is defined by kxkςX = ςkxkX .
Since W is dense in V , so V ∗ and V ∗ are dense in W ∗ . The dual space of
W ∗ ∩ V ∗ is, by Lemma 3.2.1, W + −1 V .
If u solves (3.2.3) and ξ = −2 πV Bu ∈ V ∗ , then (u , ξ ) solves the mixed
problem
(Au, Av)U + hξ, Bvi = hF, vi ∀ v ∈ H,
hη, Bui − 2 (ξ, η)V ∗ = 0 ∀ η ∈ V ∗ ,
(3.2.6)
u ∈ H, ξ ∈ V ∗ .
For this mixed problem, we have the following result
Theorem 3.2.2. The mixed problem (3.2.6) has a unique solution (u , ξ ) ∈ H × V ∗ ,
and the equivalence
ku kH + kξ kW ∗ ∩ V ∗ ' kF kH ∗
(3.2.7)
holds.
Proof. The pair (u , ξ ) solves (3.2.6) if and only if ξ = −2 πV Bu and u solves (3.2.3),
so the existence and uniqueness are established. From (3.2.4) and the first equation, we
70
get
ku kH + kξ kW ∗ . kF kH ∗ .
Taking v = u in the first equation and η = ξ in the second equation, we get the bound
on kξ k V ∗ . From the first equation, we easily get that kF kH ∗ . ku kH + kξ kW ∗ .
This theorem shows that the right space for the auxiliary variable ξ is W ∗ ∩ V ∗ .
To analyze the asymptotic behavior of the solution u , we also need to consider
the following general mixed problem.
(Au, Av)U + hξ, Bvi = hF, vi ∀ v ∈ H,
hη, Bui − 2 (ξ, η)V ∗ = hI, ηi ∀ η ∈ V ∗ ,
(3.2.8)
u ∈ H, ξ ∈ V ∗ .
here, I ∈ V . For this general mixed problem we have
Theorem 3.2.3. The mixed problem (3.2.8) has a unique solution (u , ξ ) ∈ H × V ∗ ,
and the equivalence
ku kH + kξ kW ∗ ∩ V ∗ ' kF kH ∗ + kIkW +−1 V
holds.
(3.2.9)
71
Proof. Let ζ∗0 = πV I, then hI, ηi = (ζ∗0 , η)V ∗ . The problem (3.2.8) can be reformulated
as
(Au, Av)U + hξ + −2 ζ∗0 , Bvi = hF, vi + −2 hζ∗0 , Bvi ∀ v ∈ H,
hη, Bui − 2 (ξ + −2 ζ∗0 , η)V ∗ = 0 ∀ η ∈ V ∗ ,
(3.2.10)
u ∈ H, ξ ∈ V ∗ .
This formulation is in the form of (3.2.6), therefore, we get the existence and uniqueness
of the solution from Theorem 3.2.2. In the following, C1 –C5 are constants independent
of .
From the first equation of (3.2.8), we get
kξ kW ∗ ≤ C1 (kF kH ∗ + kAu kU ).
(3.2.11)
Taking v = u and η = ξ in (3.2.8), and subtracting the second equation from the first
equation, we get
kAu k2U + 2 kξ k2V ∗ = hF, u i − hI, ξ i,
so we have
kAu k2U + 2 kξ k2V ∗ ≤ C2 (kF kH ∗ ku kH + kIkW +−1 V kξ kW ∗ ∩ V ∗ ).
(3.2.12)
From the second equation of (3.2.8) and the bound kIkV . kIkW +−1 V , we get
kBu kV ≤ C3 (2 kξ kV ∗ + kIkW +−1 V ).
(3.2.13)
72
Combining (3.2.11), (3.2.12) and (3.2.13), we get
kAu k2U + kBu k2V + 2 kξ k2V ∗ ≤ C4 (kF kH ∗ ku kH + kIkW +−1 V kAu kU
(3.2.14)
+kIkW +−1 V kξ k V ∗ + kIkW +−1 V kF kH ∗ + kIk2
W +−1 V
).
By using Cauchy’s inequality and (3.2.1), we get
).
kAu k2U + kBu k2V + 2 kξ k2V ∗ ≤ C5 (kF k2H ∗ + kIk2
W +−1 V
(3.2.15)
The upper bound of the left hand side in (3.2.9) follows from (3.2.11) and (3.2.15).
The other direction follows from the formulation (3.2.8) directly.
This result is an extension of an equivalence theorem established for the Reissner–
Mindlin plate bending model in [8].
3.3
Asymptotic behavior of the solution
When → 0, the behavior of the solution u of (3.2.3) is dramatically different
depending on whether
f |ker B 6= 0,
(3.3.1)
f |ker B = 0.
(3.3.2)
or
The asymptotic behavior needs to be discussed separately for these two cases. In the
first case, the solution blows up at the rate of O(−2 ). To fix the situation, we scale the
73
problem by assuming that F = −2 f is independent of . With this scaling, the problem
(3.2.3) or equivalently (3.2.6) can be viewed as a penalization of the constrained problem
1
(Au, Au)U − hF, ui.
u∈ker B 2
min
(3.3.3)
This constrained problem has a unique nonzero solution u0 ∈ ker B. This minimization
problem can also be written in mixed form as
(Au, Av)U + hξ, Bvi = hF, vi ∀ v ∈ H,
(3.3.4)
hη, Bui = 0 ∀ η ∈ W ∗ , u ∈ H, ξ ∈ W ∗ .
This mixed problem has a unique solution (u0 , ξ 0 ) with u0 ∈ ker B and ξ 0 ∈ W ∗ . And
we have the equivalence
ku0 kH + kξ 0 kW ∗ ' kF kH ∗ .
(3.3.5)
In the second case, the problem is essentially a singular perturbation problem.
We do not need to scale the problem. From the definition of W , we know that B is
surjective from H to W . By the closed range theorem in functional analysis, there exists
a unique ζ∗0 ∈ W ∗ such that
hf, vi = hζ∗0 , Bvi ∀ v ∈ H.
(3.3.6)
74
3.3.1
The case of surjective membrane–shear operator
We first discuss the simpler case in which the operator B : H → V is surjective.
An example of this case is the plane strain cylindrical shell problems discussed in the
last chapter. In this case, we have W = V , W ∗ = V ∗ , and
Theorem 3.3.1. Let H, U and V be Hilbert spaces, the linear operators A : H → U
bounded, and B : H → V bounded and surjective. We assume the equivalence (3.2.1)
holds, so the variational problem (3.2.3) has a unique solution u ∈ H.
If f |ker B 6= 0, we assume F = −2 f is independent of . Then
kAu − Au0 kU + −1 kBu kV . kξ 0 kV ∗ . kF kH ∗ ,
(3.3.7)
where (u0 , ξ 0 ) is the solution of the -independent problem (3.3.4), with u0 ∈ ker B,
ξ 0 ∈ V ∗ , and u0 =
6 0.
If f |ker B = 0, there exists a nonzero element ζ∗0 ∈ V ∗ , such that
hf, vi = hζ∗0 , Bvi = (ζ 0 , Bv)V ∀ v ∈ H.
Here ζ 0 ∈ V is the Riesz representation of ζ∗0 . There exists a unique u0 ∈ H, satisfying
Bu0 = ζ 0 together with the estimate
ku − u0 kH . 2 kζ 0 kV .
(3.3.8)
75
Moreover
−1 kAu − Au0 kU + −1 kBu − ζ 0 kV . kζ 0 kV = kf kH ∗ .
(3.3.9)
Proof. We prove (3.3.7) first. Under the assumption of W = V , the solutions of the
mixed problems (3.2.6) and (3.3.4) satisfy
(Au , Av)U + hξ , Bvi = hF, vi ∀ v ∈ H,
hBu , ηi − 2 (ξ , η)V ∗ = 0 ∀ η ∈ V ∗
and
(Au0 , Av)U + hξ 0 , Bvi = hF, vi ∀ v ∈ H,
hη, Bu0 i = 0 ∀ η ∈ V ∗ ,
respectively. Subtracting the second equation from the first one, and taking v = u − u0 ,
η = ξ − ξ 0 , we get
(Au − Au0 , Au − Au0 )U + 2 (ξ − ξ 0 , ξ − ξ 0 )V ∗ = − 2 (ξ 0 , ξ − ξ 0 )V ∗ .
By using Cauchy’s inequality, we get
kAu − Au0 k2U + 2 kξ − ξ 0 k2V ∗ . 2 kξ 0 k2V ∗ .
The estimate (3.3.7) then follows from the fact that ξ = −2 πV Bu .
76
Now we assume f |ker B = 0. The variational problem (3.2.3) can be written as
2 [(Au , Av)U + (Bu , Bv)V ] + (1 − 2 )(Bu , Bv)V = hf, vi
∀ v ∈ H.
(3.3.10)
By the equivalency assumption (3.2.1), the bilinear form
(u, v)H = (Au, Av)U + (Bu, Bv)V
defines an inner product on H, which is equivalent to the original inner product. With
this new inner product, the space H will be denoted by H. The condition f |ker B = 0
means that there exists a unique u0 ∈ (ker B)⊥ , the orthogonal complement of ker B in
H, such that
hf, vi = (Bu0 , Bv)V ∀ v ∈ H,
(3.3.11)
and the operator B defines an isomorphism between (ker B)⊥ and V .
From the equation (3.3.10), it is not hard to see that u ∈ (ker B)⊥ . Substituting
(3.3.11) into (3.3.10), and taking v = u − u0 , with a little algebra, we get
2 (u − u0 , u − u0 )H + (1 − 2 )(Bu − Bu0 , Bu − Bu0 )V
(3.3.12)
= 2 (Bu0 , Bu − Bu0 )V − 2 (u0 , u − u0 )H .
77
Since u and u0 both belong to (ker B)⊥ , we have ku0 kH . kBu0 kV and ku − u0 kH .
kBu − Bu0 kV . Therefore, by using Cauchy’s inequality, from (3.3.12), we get
2 (u − u0 , u − u0 )H + (1 − 2 )(Bu − Bu0 , Bu − Bu0 )V . 4 kBu0 k2V .
(3.3.13)
Therefore,
ku − u0 kH ' kBu − Bu0 kV . 2 kBu0 kV = 2 kζ 0 kV .
The estimate (3.3.9) is a consequence of the equivalence assumption (3.2.1).
To get the estimate (3.3.8), the assumption that the operator B has closed range
is crucial. Without this assumption, we can not expect the convergence of the sequence
{u } in the space H. This is a usual phenomena in singular perturbation problems and
a big trouble for numerical analysis of the general membrane–shear shells. This result is
the infinite dimensional version of the so-called Cheshire lemma.
The condition that the operator B is surjective is only satisfied by some special
problems. This condition is not met by the Reissner–Mindlin plate bending model, nor
the Koiter or Naghdi shell models. It does not apply to the spherical shell model and
the general shell model we are going to derive and justify either.
If the range of B is not closed, the space W is a proper subspace of V , so is V ∗
of W ∗ . If V ∗ is identified with V through the Riesz representation theorem, we have
the inclusions W ⊂ V ∼ V ∗ ⊂ W ∗ . We will show that in the case of f |ker B 6= 0, the
asymptotic behavior of the solution u is largely determined by the position of ξ 0 , the
Lagrange multiplier defined in (3.3.4), between V ∗ and W ∗ . The closer ξ 0 to V ∗ , the
stronger the convergence. In the best case of ξ 0 ∈ V ∗ , a convergence rate of the form
78
(3.3.7) can be obtained. In the worst case, i.e., we only have ξ 0 ∈ W ∗ , we will prove a
convergence, but without convergence rate.
In the case of f |ker B = 0, we must require ζ∗0 , the equivalent representation of f
in W ∗ defined in (3.3.6), to be in the smaller space V ∗ . Then, the asymptotic behavior
of u is determined by the position of ζ 0 , the Riesz representation of ζ∗0 in the space V ,
between W and V .
3.3.2
The case of flexural domination
In this subsection, we discuss the case of f |ker B 6= 0 without the assumption that
B is surjective. In this case we need to rescale the problem by assuming that F = −2 f
is independent of . We have
Theorem 3.3.2. Let (u , ξ ) be the solution of (3.2.6), and (u0 , ξ 0 ) be the solution of
(3.3.4), then
kAu − Au0 kU + −1 kBu kV . kξ 0 k−1 W ∗ +V ∗
(3.3.14)
Proof. Subtracting (3.3.4) from (3.2.6), we get
(A(u − u0 ), Av)U + hξ − ξ 0 , Bvi = 0 ∀ v ∈ H,
(3.3.15)
hB(u − u0 ), ηi − 2 (ξ , η)V ∗ = 0 ∀ η ∈ V ∗ .
Taking v = u − u0 and η = ξ , and writing the second equation as
hB(u − u0 ), ξ − ξ 0 i + hB(u − u0 ), ξ 0 i − 2 (ξ , ξ )V ∗ = 0,
79
together with the first equation
(A(u − u0 ), A(u − u0 ))U + hξ − ξ 0 , B(u − u0 )i = 0,
we get
(A(u − u0 ), A(u − u0 ))U + 2 (ξ , ξ )V ∗ = hB(u − u0 ), ξ 0 i.
(3.3.16)
By the definition (3.2.5) of the norm of W , and the equivalence assumption (3.2.1), we
have
kB(u − u0 )kW . kA(u − u0 )kU + kB(u − u0 )kV ,
and so
kB(u − u0 )k W . kA(u − u0 )kU + kB(u − u0 )kV .
Therefore, we have the estimate
|hB(u − u0 ), ξ 0 i| . [ kA(u − u0 )kU + kB(u − u0 )kV ]kξ 0 k−1 W ∗ +V ∗ .
(3.3.17)
Recalling that ξ = πV −2 Bu and Bu0 = 0, combining (3.3.16) and (3.3.17)
and using Cauchy’s inequality, we obtain
kA(u − u0 )k2U + −2 kBu k2V . 2 kξ 0 k2−1
W ∗ +V ∗
.
80
The desired result then follows.
The K-functional on the Hilbert couple [W ∗ , V ∗ ] is given by
K(, ξ 0 , [W ∗ , V ∗ ]) = kξ 0 k−1 W ∗ +V ∗ ,
see [9]. According to the definition of interpolation spaces based on the K-functional,
|K(, ξ 0 , [W ∗ , V ∗ ])| . Cθ,q θ kξ 0 k[V ∗ ,W ∗ ]
1−θ,q
.
If ξ 0 is further assumed to belong to the interpolation space [V ∗ , W ∗ ]1−θ,q , for some
0 < θ < 1 and 1 ≤ q ≤ ∞, or 0 ≤ θ ≤ 1 and 1 < q < ∞, we have
kAu − Au0 kU + −1 kBu kV . θ kξ 0 k[V ∗ ,W ∗ ]
1−θ,q
.
(3.3.18)
In particular, if
ξ0 ∈ V ∗,
(3.3.19)
we can take θ = 1 and obtain the stronger result
kAu − Au0 kU + −1 kBu kV . kξ 0 kV ∗ .
(3.3.20)
The estimate (3.3.20) is an extension of a convergence theorem of solution of
the Reissner–Mindlin plate bending model to that of the Kirchhoff–Love plate bending
model in [4].
81
We will prove that the convergence rate of our 2D shell model solution to the 3D
shell solution in the relative energy norm is crucially related to this “regularity index” θ
of the Lagrange multiplier ξ 0 .
The condition (3.3.19) can be verified for the Reissner–Mindlin plate bending
model, if the plate is totally clamped, and the plate boundary and loading function are
smooth enough so that the H 3 regularity of the Kirchhoff–Love model solution holds,
see [8]. For partially clamped plates and arbitrary shells, this index needs to be carefully
evaluated.
If we know nothing more than the minimum regularity of the Lagrange multiplier
ξ 0 ∈ W ∗ , then we must choose θ = 0. The estimate (3.3.18) does not provide any useful
information. The following theorem will be used to prove the convergence, but without
a convergence rate, of the flexural shell model.
Theorem 3.3.3. Let u be the solution of (3.2.6) and u0 be the solution of (3.3.4). We
have the convergence result
lim [kA(u − u0 )kU + −1 kBu kV ] = 0.
→0
Proof. Taking v = u in the equation (3.2.3), we get
(Au , Au )U + −2 (Bu , Bu )V = hF, u i.
(3.3.21)
82
From this equation we see that there exists a constant C independent of , such
that
ku kH + k −1 Bu kV ≤ C.
Since bounded sets in a Hilbert space are weakly compact, there exists a subsequence,
n → 0, an element ũ ∈ H, and an element p ∈ V , such that
un * ũ in H,
n
−1
n Bu * p in V.
Since Bun * B ũ in V , and Bun → 0 in V , we have B ũ = 0, so ũ ∈ ker B. Note that
we also have Aun * Aũ, so we have
(Aũ, Av)U = hF, vi ∀ v ∈ ker B,
so ũ = u0 , the solution of (3.3.3). Therefore the whole sequence {u } weakly converges
to u0 in H.
Consider the identity
(Au − Au0 , Au − Au0 )U + (−1 Bu − p, −1 Bu − p)V
= (Au , Au )U + −2 (Bu , Bu )V + (Au0 , Au0 − 2Au )U + (p, p − 2 −1 Bu )V
= hF, u i + (Au0 , Au0 − 2Au )U + (p, p − 2 −1 Bu )V . (3.3.22)
83
For the above subsequence, the right hand side, as a sequence of numbers, converges to
hF, u0 i − (Au0 , Au0 )U − (p, p)V = −(p, p)V ,
while the left hand side of (3.3.22) is nonnegative, so we must have p = 0. Therefore
the whole sequence −1 Bu weakly converges to zero. The desired strong convergence
follows from the same identity.
This theorem is an extension of, and its proof was adapted from [4] for Reissner–Mindlin
plate bending, [21] for flexural Naghdi shell, and [15] for flexural Koiter shell problems.
This theorem shows that if f |ker B 6= 0, the problem is bending or flexural dominated in the sense of
(Bu , Bu )V
→ 0 ( → 0).
2 (Au , Au )U
3.3.3
The case of membrane–shear domination
If f |ker B = 0, the solution of the limiting problem (3.3.3) will be zero. If we
still assume F = −2 f is independent of , the above estimate only gives the following
convergence.
kAu kU + −1 kBu kV → 0 ( → 0).
(3.3.23)
This convergence will be useful when we analyze the relationship between our theory
and other shell theories. It is also needed to resolve the effect of the higher order term
2 f1 in the loading functional. Otherwise, it hardly tells us more than that the solution
u converges to zero, and fails to fully capture the asymptotic behavior of the solution.
To get a good grasp of the asymptotic behavior of the solution, we will assume that f is
84
independent of in this case. In this case, there exists a unique ζ∗0 ∈ W ∗ such that
hf, vi = hζ∗0 , Bvi ∀ v ∈ H.
Equivalently,
hF, vi = h−2 ζ∗0 , Bvi ∀ v ∈ H.
Without further assumption on ζ∗0 , we can not get any useful results for our model
justification. We will derive an estimate under the assumption
ζ∗0 ∈ V ∗ ,
(3.3.24)
so ζ 0 = iV ζ∗0 is well-defined. This condition does exclude some physically meaningful
shell problems. However, if this condition is not satisfied, the 2D model solution, very
likely, does not approximate the 3D elasticity solution in the energy norm.
The mixed problem (3.2.6) may be written as
(Au, Av)U + hξ, Bvi = h−2 ζ∗0 , Bvi ∀ v ∈ H,
hη, Bui − 2 (ξ, η)V ∗ = 0 ∀ η ∈ V ∗ ,
u ∈ H, ξ ∈ V ∗ .
85
Under the assumption (3.3.24), this problem can be rewritten as
(Au, Av)U + hξ − −2 ζ∗0 , Bvi = 0 ∀ v ∈ H,
hη, Bui − 2 (ξ − −2 ζ∗0 , η)V ∗ = (ζ∗0 , η)V ∗ = hζ 0 , ηi
(3.3.25)
∀ η ∈ V ∗ , u ∈ H, ξ ∈ V ∗ .
This formulation is in the form of our general mixed problem (3.2.8). Therefore, by
Theorem 3.2.3, we have the equivalence
ku kH + kξ − −2 ζ∗0 kW ∗ ∩ V ∗ ' kζ 0 kW +−1 V .
(3.3.26)
Recalling that ξ = −2 πV Bu , we get the equivalence
ku kH + −2 kπV Bu − ζ∗0 kW ∗ ∩ V ∗ ' kζ 0 kW +−1 V .
Therefore,
ku kH + −2 kπV Bu − ζ∗0 kW ∗ + −1 kBu − ζ 0 kV ' kζ 0 kW +−1 V .
In particular, we have proved the following theorem.
(3.3.27)
86
Theorem 3.3.4. In the case of f |ker B = 0, and under the assumption ζ∗0 ∈ V ∗ , we
have the following estimates:
kAu kU + kBu − ζ 0 kV . kζ 0 kW +−1 V ,
(3.3.28)
kπV Bu − ζ∗0 kW ∗ . 2 kζ 0 kW +−1 V .
In terms of the K-functional, we have
kζ 0 kW +−1 V = K(, ζ 0 , [V, W ]).
If ζ 0 belongs to the interpolation space [W, V ]1−θ,q for some 0 < θ < 1 and 1 ≤ q ≤ ∞,
or 0 ≤ θ ≤ 1 and 1 < q < ∞, we have
kAu kU + kBu − ζ 0 kV . θ kζ 0 k[W,V ]
1−θ,q
.
(3.3.29)
In particular, if ζ 0 ∈ W , we can take θ = 1 and obtain
kAu kU + kBu − ζ 0 kV . kζ 0 kW .
(3.3.30)
The “regularity index” θ of ζ 0 , i.e., the largest θ such that ζ 0 ∈ [W, V ]1−θ,q ,
which will determine the convergence rate of the shell model in the relative energy norm,
can be attributed to the regularity of the shell data, but generally it is hard to interpret
in terms of smoothness in the Sobolev sense. For a totally clamped elliptic shell, we can
show that θ = 1/6 under smoothness assumptions on the shell boundary and loading
87
functions in the usual sense. For the shear dominated Reissner–Mindlin plate bending,
reasonable assumptions on the smoothness of the loading functions will lead to θ = 1/2.
If we only have the minimum regularity assumption ζ 0 ∈ V , we just have θ = 0,
and the estimate (3.3.29) reduces to
kAu kU + kBu − ζ 0 kV . kζ 0 kV ,
(3.3.31)
which does not tell anything useful. We can construct example to show that this estimation is optimal. Due to the independence of f , we have the strong convergence
stated in the next theorem. This convergence will be used to prove the convergence of
the model, although without a convergence rate.
Theorem 3.3.5. If f |ker B = 0, and its representative ζ∗0 ∈ V ∗ , we have the strong
convergence
lim [ kAu kU + kBu − ζ 0 kV ] = 0.
→0
Proof. From (3.3.31), we see that there exist a constant C independent of , such that
k Au kU ≤ C,
kBu kV ≤ C.
Therefore, there exist a subsequence, n → 0, an element p ∈ U , and an element v 0 ∈ V ,
such that n Aun * p in U , and Bun * v 0 in V . Since
2n (Aun , Av)U + (Bun , Bv)V = (ζ 0 , Bv)V ∀v ∈ V,
88
we have Bun * ζ 0 in V, therefore v 0 = ζ 0 .
The following identity can be verified:
( Au − p, Au − p)U + (Bu − ζ 0 , Bu − ζ 0 )V
= 2 (Au , Au )U + (Bu , Bu )V + (p, p − 2 Au )U + (ζ 0 , ζ 0 − 2Bu )V
= (ζ 0 , Bu )V + (p, p − 2 Au )U + (ζ 0 , ζ 0 − 2Bu )V . (3.3.32)
When applied to the subsequence {un }, the right hand side of this identity converges
to
(ζ 0 , ζ 0 )V − (p, p)U − (ζ 0 , ζ 0 )V = −(p, p)U .
Since the left-hand side is nonnegative, we must have p = 0. Therefore, the whole
sequence Au weakly converges to zero, and the whole sequence Bu weakly converges
to ζ 0 . The strong convergence follows from the above identity.
This proof was adapted from [36] for singular perturbation problems, [21] for
membrane Koiter shell, and [15] for membrane Naghdi shell.
Under the condition of this theorem, the problem is membrane–shear dominated
in the sense of
2 (Au , Au )U
→ 0 ( → 0).
(Bu , Bu )V
The above analysis shows that if f |ker B = 0 and ζ∗0 ∈ V ∗ , or more informatively,
ζ 0 ∈ [W, V ], we have membrane–shear domination. If ζ∗0 does not belong to V ∗ , the
behavior of the solution can be very complicated. Usually, there is no membrane-shear
89
domination, but rather, the flexural energy 2 (Au , Au )U might be comparable to the
membrane–shear energy (Bu , Bu )V , see [12] and [41], although the geometry of its
middle surface and the type of the boundary conditions may classify a shell as a membrane shell. For example, a partially clamped elliptic shell may behave this way even
for infinitely differentiable loading functions. In this case, the limiting membrane shell
model has no solution. Although our model provides a solution, we are not able to justify
it.
The following corollary to Theorems 3.3.4 and 3.3.5 will be used when we construct
corrections for the transverse deflection, which are necessary for the convergence of the
shell model in the relative energy norm.
Theorem 3.3.6. Let ω ⊂ R2 be a bounded, connected open domain whose boundary is
1 is a subspace of H 1 whose
partitioned as ∂ω = ∂D ω ∪ ∂T ω. The function space HD
elements vanish on ∂D ω. The variational problem
1,
2 (∇u , ∇v)L2 + (u , v)L2 = hf, vi ∀v ∈ HD
∼
1
u ∈ H D
(3.3.33)
1 for any f ∈ H 1 ∗ . If f ∈ L , we have the estimate
has a unique solution u ∈ HD
2
D
k∇u kL2 + ku − f kL2 . kf kH 1 +−1 L .
∼
D
2
(3.3.34)
If f ∈ H 1 , the standard cut-off argument gives
k∇u kL2 + ku − f kL2 . 1/2 kf kH 1 .
∼
(3.3.35)
90
If we assume that the interpolation norm kf k[H 1 ,L ]
is finite for some θ ∈ (0, 1)
D 2 1−θ,q
and q ∈ [1, ∞], or θ ∈ [0, 1] and q ∈ (1, ∞), we have
.
k∇u kL2 + ku − f kL2 . θ kf k[H 1 ,L ]
∼
D 2 1−θ,q
(3.3.36)
1 , we have
In particular, if f ∈ HD
k∇u kL2 + ku − f kL2 . kf kH 1 .
∼
(3.3.37)
If we only know that f ∈ L2 , the strong convergence
lim [ k∇u kL2 + ku − f kL2 ] = 0
→0
∼
(3.3.38)
holds.
1,
Proof. The conclusions follow from the above theorems by letting H = HD
U = L2 , V = L2 , A = ∇, and B = identity.
∼
A direct proof of (3.3.35) can be found in [2].
3.4
Parameter-dependent loading functional
In this section, we discuss the behavior of the solution of the variational problem
2 (Au, Av)U + (Bu, Bv)V = hf0 + 2 f1 , vi,
(3.4.1)
u ∈ H,
∀ v ∈ H,
91
in which both f0 and f1 are independent of , and f0 6= 0. This is a problem abstracted
from our shell models. This form of the resultant loading functional is a consequence
of our assumption on the loading functions. To grasp the behavior of solution of the
problem with such -dependent loading functionals, we apply the above theory to the
problems whose right hand sides are f0 and 2 f1 respectively. The desired behavior will
be obtained by superposition.
Let f = f0 in the equations and theorems in the previous two sections, and
consider the problem
2 (Au1 , Av)U + (Bu1 , Bv)V = 2 hf1 , vi,
(3.4.2)
u1 ∈ H,
∀ v ∈ H,
which is due to the higher order term in the loading functional. This problem has a
unique solution u1 , and by Theorem 3.3.3, we have
lim [kA(u1 − u01 )kU + −1 kBu1 kV ] = 0,
→0
(3.4.3)
where u01 ∈ H is defined as the solution of (3.3.3) or (3.3.4) with F replaced by f1 . Note
that u01 may be zero or nonzero depending on whether f1 |ker B = 0 or not.
We will see that the problem should be classified by the leading term f0 , and we
discuss the problem separately for whether or not f0 |ker B 6= 0.
If f0 |ker B 6= 0, we need to scale the loading functionals f0 and 2 f1 simultaneously. The solution of (3.2.2) will be given by ũ = u + 2 u1 with u and u1 the
92
solutions of (3.2.6) and (3.4.2) respectively. Under the condition of Theorem 3.3.2, by
(3.3.18) together with (3.4.3) we get the estimate
kAũ − Au0 kU + −1 kB ũ kV
. kAu − Au0 kU + −1 kBu kV + 2 (kAu1 kU + −1 kBu1 kV )
. θ kξ 0 k[V ∗ ,W ∗ ]
1−θ,q
+ O(2 ). (3.4.4)
Under the condition of Theorem 3.3.3, we have
kAũ − Au0 kU + −1 kB ũ kV . o(1) + O(2 ).
(3.4.5)
Therefore, in the case of flexural shells, adding the higher order term 2 f1 to the right
hand side does not disturb the asymptotic behavior of the solution of (3.2.2) determined
by the leading term.
If f |ker B = 0, there is no need to scale the loading functional and the solution of
(3.2.2) is given by ũ = u + u1 with u and u1 defined as solutions of (3.2.3) and (3.4.2)
respectively. Under the condition of Theorem 3.3.4, corresponding to the convergence
(3.3.29), by using (3.4.3), we have
kAũ kU + kB ũ − ζ 0 kV
. kAu kU + kBu − ζ 0 kV + (kAu1 kU + −1 kBu1 kV )
. θ kζ 0 k[W,V ]
1−θ,q
+ O()[o()], (3.4.6)
93
if u01 =
6 0[= 0]. Under the condition of Theorem 3.3.5, we have
kAũ kU + kB ũ − ζ 0 kV . o(1) + O()[o()],
(3.4.7)
if u01 =
6 0[= 0]. Therefore, the higher order term 2 f1 does not affect the asymptotic
behaviors of the solution of (3.2.2), as described by Theorems 3.3.4 and 3.3.5.
If the range of B is closed, or in the situation of Theorem 3.3.1, the additional
term 2 f1 does not affect the asymptotic behaviors described by (3.3.7) and (3.3.9).
But the stronger convergence (3.3.8) will be affected, especially when f |ker B = 0 and
f1 |ker B 6= 0. In this case the contribution to the solution from 2 f1 will be finite, by
correcting the limit, we will get a convergence in the form of (3.3.8) while the convergence
rate needs to be reduced from 2 to , see (2.5.4) and (2.5.7).
In Chapter 7, we will discuss the the model under the usual assumption on the
loading functions. For that purpose, we need to consider the problem
2 (Au, Av)U + (Bu, Bv)V = hf0 + f1 + 2 f2 + 3 f3 , vi,
(3.4.8)
u ∈ H,
∀ v ∈ H,
with f0 , f1 , f2 , and f3 independent of , and f0 6= 0. The theory can be applied to
problems of the form (3.2.3), with right hand sides f0 , f1 , 2 f2 , and 3 f3 , respectively.
The desired behavior will be obtained by superposition. Since we will not discuss the
convergence rate of other shell theories in details in this thesis, we will not list the results
94
corresponding to (3.4.4) and (3.4.6). We still denote the solution of (3.4.8) by ũ . The
following convergence results can be obtained.
If f0 |ker B 6= 0, we have
kAũ − Au0 kU + −1 kB ũ kV . o(1).
(3.4.9)
If f0 |ker B = 0, f1 |ker B = 0, under the condition of theorem 3.3.5 we have
kAũ kU + kB ũ − ζ 0 kV . o(1).
(3.4.10)
The asymptotic behavior of solution of (3.4.1) was not harmed by adding f1 + 2 f2 +
3 f3 to the loading functional.
If f0 |ker B = 0 but f1 |ker B 6= 0, we only have
kAũ kU + kB ũ − ζ 0 kV . O(1).
(3.4.11)
In this case, the expected membrane-shear dominated asymptotic behavior described by
Theorem 3.3.5 was severely affected by adding the higher order term f1 to the loading
functional. This is a rare situation in which the leading term f0 puts the problem in the
category of membrane-shear shells, while the higher order term f1 draws it into the
category of flexural shell. In the sum, neither of them can dominate.
95
3.5
Classification
For the abstract variational problem
2 (Au , Av)U + (Bu , Bv)V = hf0 + 2 f1 , vi,
u ∈ H,
∀ v ∈ H,
the two estimates (3.4.4) and (3.4.5) show that if f0 |ker B 6= 0, the flexural energy
2 (Au , Au )U dominates. In this case, the shell problem will be called a flexural shell.
The two estimates (3.4.6) and (3.4.7) show that when f0 |ker B = 0 and its representation ζ∗0 ∈ V ∗ , the membrane–shear energy (Bu , Bu )V dominates. In this case,
the shell problem will be called a membrane–shear shell. A membrane–shear shell will
be called a first kind membrane–shear shell or stiff membrane–shear shell if ker B = 0. If
ker B 6= 0 but f0 |ker B = 0, the shell will be called a second kind membrane–shear shell.
We will justify the shell model in both of the above cases. If f0 |ker B = 0, but ζ∗0
does not belong to V ∗ , the shell model can not be justified.
96
Chapter 4
Three-dimensional shells
In this chapter, we briefly review the linearized 3D elasticity theory for a thin
elastic shell in the curvilinear coordinates and recall all the materials from the differential geometry of surfaces that will be necessary for the shell analyses. Special curvilinear
coordinates on 3D shells, which are attached to coordinates on the middle surfaces, will
be defined. Rescaled stress components, rescaled applied force components, and rescaled
displacement components will be introduced. In terms of the rescaled components, the
linearized elasticity equations have a noticeably simpler form, and calculations can be
substantially simplified. We also recall the two energies principle that will be the fundamental tool for our justification of the spherical shell model.
4.1
Curvilinear coordinates on a shell
Let ω ⊂ R2 be a bounded connected open domain, whose boundary ∂ω is smooth.
We use x = (x1 , x2 ) to denote the Cartesian coordinates of a generic point in ω̄. A
∼
surface S ⊂ R3 is defined as the image of the set ω̄ through a mapping φ from ω̄ to
R3 . We assume that the mapping is injective and fairly smooth. The boundary of S is
γ = φ(∂ω). The pair of numbers x = (x1 , x2 ) then furnishes the curvilinear coordinates
∼
on the surface S. We assume that at any point on the surface, along the coordinate lines,
the two tangential vectors aα = ∂φ/∂xα are linearly independent. The unit vector a3
97
that is normal to the surface can be expressed as
a3 =
a1 × a2
.
|a1 × a2 |
At any point on the surface, the three vectors ai furnish the covariant basis. The
contravariant basis ai is defined by the relations aα · aβ = δβα and a3 = a3 , in which
δβα is the Kronecker delta. It is obvious that aα are also tangent to the surface.
The first fundamental form on the surface, or the metric tensor, aαβ is defined
by aαβ = aα · aβ , which is symmetric positive definite. The contravariant components
of the metric tensor are given by aαβ = aα · aβ .
The second fundamental form, or the curvature tensor, bαβ is defined by bαβ =
αγ
a3 · ∂β aα , which is also symmetric. The mixed curvature tensor is bα
β = a bγβ . The
γ
tensor cαβ = bα bγβ is called the third fundamental form, which is also symmetric.
The trace and determinant of the mixed curvature tensor bα
β (as a matrix) are
intrinsic quantities of the surface which are independent of the coordinates. They are
the mean curvature and Gauss curvature respectively, denoted by
1
H = (b11 + b22 )
2
and K = b11 b22 − b12 b21 .
The three fundamental forms and the two curvatures are connected by the identity
Kaαβ − 2Hbαβ + cαβ = 0.
98
Expressed in mixed components of the tensors, this identity easily follows from the
Hamilton–Cayley theorem in matrix analysis.
γ
γ
The Christoffel symbols Γαβ are defined by Γαβ = aγ ·∂β aα , which are symmetric
γ
γ
with respect to the subscripts, i.e., Γαβ = Γβα .
The shell with middle surface S and thickness 2 , is a 3D elastic body occupying
the domain Ω ⊂ R3 , which is the image of the plate ω = ω̄ × [− , ] through the
mapping Φ:
Φ(x1 , x2 , t) = φ(x1 , x2 ) + ta3 ,
(x1 , x2 ) ∈ ω̄,
t ∈ [− , ].
We assume that is small enough so that Φ is injective. The triple of numbers (x1 , x2 , t)
furnishes the curvilinear coordinates on the shell Ω . We may use t = x3 exchangeably
for convenience. Corresponding to these curvilinear coordinates, the covariant basis
vectors at any point in Ω are defined by
g i (x1 , x2 , x3 ) =
∂Φ(x1 , x2 , x3 )
.
∂xi
The 3D second order tensor gij = g i · g j is called the covariant metric tensor, whose
determinant is denoted by g = det(gij ). The contravariant metric tensor g ij is defined
as the inverse of gij as a matrix, so, g ik gkj = δji . The triple of vectors g i = g ij g j
furnishes the contravariant basis. Note that g i · g j = δji .
A vector field v can be given in terms of its covariant components vi or contravariant components v i through the relation v = vi g i = v i g i . A tensor field σ can be
99
given in terms of its contravariant components σ ij , covariant components σij , or mixed
components σji through the relations
σ = σ ij g i ⊗ g j = σij g i ⊗ g j = σji g i ⊗ g j .
For brevity, we will use notations like v = vi = v i and σ = σ ij = σij = σji . The
covariant components of a tensor will be called a covariant tensor, etc.
k
The Christoffel symbols are defined by Γ∗k
ij = g ·∂j g i . The superscript ∗ is added
to indicate the difference from the Christoffel symbols on the middle surface. The indices
of all tensors and the Christoffel symbols can be raised or lowered by multiplication and
contraction with the contravariant or covariant metric tensors.
For any vector or tensor defined on the shell Ω , we can define its covariant
and contravariant derivatives, which themselves are tensors of higher orders. We use
double vertical bar to denote the derivatives on the 3D shell. For example, the covariant
derivative of the stress tensor σ ij is a third order 3D tensor, whose mixed components
are given by
∗j
mj + Γ σ in .
σ ij kk = ∂k σ ij + Γ∗i
km σ
kn
The row divergence of the stress tensor σ ij is a vector whose contravariant components are obtained from a contraction of the above third order tensor.
∗j
mj + Γ σ in .
div σ = σ ij kj = ∂j σ ij + Γ∗i
jm σ
jn
(4.1.1)
100
The covariant derivative of a vector v = vi is a second order tensor with covariant
components
vikj = ∂j vi − Γ∗k
ij vk .
In terms of the contravariant components v i , the mixed components of the covariant
derivative of v can be expressed as
k
v i kj = ∂j v i + Γ∗i
kj v .
Note that for any vector field or tensor field defined on the shell Ω , its components
can be viewed as functions defined on the coordinate domain ω . Sometimes, we may
slightly abuse notations by discarding the difference between functions defined on Ω
and ω . The distinction should be clear from the context.
On the middle surface S, we can define the covariant and contravariant derivatives
of any 2D vectors or tensors. The derivative will be denoted by a single vertical bar. A
2D tensor can be viewed as the restriction on the middle surface of a 3D tensor with zero
non-tangential components. On the middle surface, the tangential part of the derivative
of this 3D tensor is defined as the derivative of the 2D tensor. For example, on the surface
S, the covariant derivative of the second order tensor σ = σ αβ is defined in terms of its
∼
mixed components by the first equation in (4.1.2) below. The covariant derivative of the
second order tensor τ = τβα is given by the second equation. The covariant derivative
∼
of the vector field u = uα aα = uβ aβ is given in terms of its covariant components and
∼
101
mixed components by the last two equations respectively.
β
λβ + Γ σ ατ ,
σ αβ |γ = ∂γ σ αβ + Γα
γτ
γλ σ
τ
γ
γ
γ
γ
= ∂β τα + Γλβ ταλ − Γταβ ττ ,
α|β
γ
uα|β = ∂β uα − Γαβ uγ ,
(4.1.2)
γ
uα |β = ∂β uα + Γα
γβ u .
The mixed components of the covariant derivative of the curvature tensor b
γ
γ
γ
τ
∂β bα + Γλβ bλ
α − Γαβ bτ is symmetric about the subscripts, i.e., b
γ
=
α|β
γ
γ
= b . This is the
α|β
β|α
Codazzi–Mainardi identity, which follows from the second equation in (4.1.6) below. It
actually is a consequence of the assumption that the surface S can be embedded in the
Euclidean 3 space.
We formally define the surface covariant derivatives for the tangential parts of 3D
tensors defined on the shell Ω , by the same formulae (4.1.2), and denote them by the
same notations. For example, if τ = τ ij (x , t) is a tensor field defined on the shell Ω ,
∼
for any given t0 ∈ [− , ], τ αβ (x , t0 ) can be viewed as the contravariant components of
∼
a 2D tensor defined on the middle surface. We will define τ αβ |γ at any point (x , t0 ) by
∼
the formula
β
λβ + Γ τ αλ .
τ αβ |γ = ∂γ τ αβ + Γα
γλ τ
γλ
It is important to note that the derivatives denoted by a single vertical bar are always
taken with respect to the metric on the middle surface. More specifically, the Christoffel
symbols in the right-hand side of the above equation are those defined on the middle
surface.
102
Product rules for differentiations like
(σ ij uj )kk = σ ij kk uj + σ ij ujkk ,
(4.1.3)
(σ αλ uλ )|β = σ αλ |β uλ + σ αλ uλ|β ,
are, of course, always valid.
The following Green’s theorem, or divergence theorem, on the surface S will be
frequently used. Let n = nα aα be the unit outward normal in the surface S to its
boundary γ, then
Z
S
uα |α dS =
Z
γ
uα nα dγ
(4.1.4)
holds for any vector field u = uα aα defined on S. In the above equation, the left hand
side integral is taken with respect to the surface area element and the right hand side
integral is taken with respect to the arc length of the boundary curve γ.
Our ultimate goal is to approximate the 3D problem defined on the shell Ω
by a 2D problem defined on the middle surface S, so it is indispensable to make the
dependence of various quantities on the transverse coordinate t as explicit as possible.
). The dependence of this tensor valued function,
We set µα
, t) = δβα − tbα
β (x
β (x
∼
∼
and of all the functions that will be introduced later, on the coordinates (x , t) will not
∼
be indicated explicitly in the following, but should be clear from the context. We denote
α
2
the determinant of µα
β by ρ = det(µβ ) = 1 − 2Ht + Kt .
Let a = det(aαβ ). Then the area element on S is
the shell Ω is
√
gdx dt. The relation
∼
√
√
g = ρ a holds.
√
adx . The volume element in
∼
103
γ
α
α
The mixed tensor ζβα is defined as the inverse of µα
β (as a matrix), so ζγ µβ = δβ .
From Cramer’s rule, we have the expression
ζβα =
1 αγ λ
δ µ .
ρ βλ γ
αγ
Here δβλ = αγ βλ is the generalized Kronecker delta. The -systems on the surface S
√
√
are defined by 11 = 22 = 0, 12 = − 21 = a, and 11 = 22 = 0, 12 = − 21 = 1/ a.
αγ
λ
We define the mixed tensor dα
β = δβλ bγ , which is the cofactor of the mixed curvature
α
tensor. Then we have ρζβα = δβα −tdα
β . Note that ρζβ is a linear function in the transverse
coordinate t. This simple observation will play an important role in our model derivation
for general shells.
α
Between the curvature tensor bα
β and the tensor dβ , the following relations hold:
α
λ
dα
λ bβ = Kδβ ,
α
α
dα
β + bβ = 2Hδβ .
The basis vectors and metric tensor at any point in the 3D shell are related to
corresponding quantities at the projected point on the middle surface by the following
equations:
γ
g α = µα aγ ,
g α = ζγα aγ ,
g 3 = g 3 = a3 = a3 ,
(4.1.5)
γ
2
gαβ = µα µλ
β aγλ = aαβ − 2tbαβ + t cαβ , gα3 = g3α = 0, g33 = 1.
104
Some important relations for the Christoffel symbols are
∗γ
∗3
Γ33 = Γ∗3
3α = Γα3 = 0,
∗γ
∗γ
γ
γ
Γαβ = Γβα = Γαβ − tζλ bλ
α|β ,
Γ∗3
αβ = bαβ − tcαβ ,
(4.1.6)
α λ
Γ∗α
3β = −ζλ bβ ,
especially,
α
Γ∗α
3β |t=0 = −bβ .
Γ∗3
αβ |t=0 = bαβ ,
The proofs of these relations are direct applications of the definition of the Christoffel
symbols. We just prove the second equation which we have not found in the literature,
but is necessary for us.
By the definitions of Christoffel symbols on both the middle
surface S and the 3D shell Ω , and the relations (4.1.5), we have
γ τ
∗γ
γ
λ
λ
Γαβ = g γ · ∂β g α = ζτ aτ · ∂β (µλ
α aλ ) = ζτ a · (∂β µα aλ + µα ∂β aλ )
γ
γ
σ λ
λ
λ τ
= ζλ (∂β µλ
α + Γτ β µα ) = ζλ (µα|β + Γαβ µσ )
γ
γ
γ
γ
λ
= Γαβ + ζλ µλ
α|β = Γαβ − tζλ bα|β .
Let the boundary of ω be divided to distinct parts as ∂ω = ∂D ω ∪ ∂T ω, with
∂D ω ∩ ∂T ω = ∅, giving the clamping and traction parts of the shell lateral surface. The
boundary of the middle surface S will be correspondingly divided as γ = γT ∪ γD . The
boundary of the shell Ω is composed of the upper and lower surfaces Γ± = Φ(ω ×{± })
105
where the shell is subjected to surface tractions, the clamping lateral surface ΓD =
Φ(∂D ω × [− , ]) where the shell is clamped (the cross hatched part of the lateral surface
in Figure 4.1), and the remaining part of the lateral surface ΓT = Φ(∂T ω×[− , ]), where
the shell is under traction or free.
t
Ω
x2
x1
Φ
φ
2
ω
Fig. 4.1. A shell and its coordinate domain
The unit outer normal on Γ+ is given by n+ = g 3 . The unit outer normal on
Γ− is n− = −g 3 . On ΓT , at the point Φ(x , t) ∈ ΓT (x ∈ ∂T ω), we denote the unit
∼
∼
outer normal by n∗ which is obviously parallel to the middle surface, so it can be
expressed as n∗ = n∗α g α . Here the superscript ∗ was added to indicate the dependence
of the components on t. Let n = nα aα be the unit outer normal in the surface S to
its boundary γT . And let x (s) be the arc length parameterization of ∂T ω, and ẋ (s) the
∼
∼
106
unit tangent vector to the curve ∂T ω at the point x (s), then it can be shown that
∼
ρ
n∗α = − αβ ẋβ (s).
η
As ρ measures the transverse volume variation of the shell body, the function
η(x , t) measures the transverse area variation of the shell lateral surface. It is given by
∼
q
gαβ (x (s), t)ẋα (s)ẋβ (s)
∼
∀ x (s) ∈ ∂T ω,
η(x (s), t) = q
∼
∼
aαβ (x (s))ẋα (s)ẋβ (s)
∼
so we have
n∗α =
4.2
ρ
nα .
η
(4.1.7)
Linearized elasticity theory
In the context of the linearized elasticity, the deformation and stress distribution
in an elastic shell arising in response to the applied forces and boundary conditions are
determined by the geometric equation (4.2.1), the constitutive equation (4.2.2), the equilibrium equation (4.2.3), and traction (4.2.4) and clamping (4.2.5) boundary conditions
on the shell surface.
Let the surface force densities on Γ± be p± = pi± g i , the surface force density
on ΓT be pT = piT g i , and the body force density be q = q i g i .
covariant basis vectors at the relevant point.
Note that g i are the
107
Let v be the displacement vector field, χij the strain tensor field, and σ ij the
stress tensor field. The displacement-strain relation, or geometric equation, is
1
χij = (vikj + vjki ).
2
(4.2.1)
The constitutive equation, which connects stress to strain, is
σ ij = C ijkl χkl
or χij = Aijkl σ kl ,
(4.2.2)
where the 3D fourth order tensors C ijkl and Aijkl are the elasticity tensor and the
compliance tensor. They are given by
C ijkl = 2µg ik g jl + λg ij g kl
λ
1
[gik gjl −
g g ],
and Aijkl =
2µ
2µ + 3λ ij kl
respectively. The equilibrium equation, expressed in terms of the tensor and vector
components, is
σ ij kj + q i = 0.
(4.2.3)
On Γ± and ΓT , the surface force condition, expressed in terms of the contravariant stress
components, is
j
σ 3j = p+ on Γ+ ;
j
σ 3j = −p− on Γ− ;
j
σ jα n∗α = pT on ΓT .
(4.2.4)
108
On ΓD the shell is clamped, so the displacement vanishes, and the condition is
vi = 0 on ΓD .
(4.2.5)
The theory of linearized 3D elasticity says that the system of equations (4.2.1),
(4.2.2), (4.2.3), together with the boundary conditions (4.2.4) and (4.2.5) uniquely determine the displacement v ∗ = vi∗ and the stress σ ∗ = σ ∗ij distributions over the
loaded shell arising in response to the applied body force, surface force, and clamping
boundary condition. The displacement v ∗ can be determined as the unique solution of
the weak form of the 3D elasticity equations:
Z
Ω
C ijkl χkl (v)χij (u) =
Z
Ω
v ∈ H 1D (ω ),
q i ui +
Z
Γ±
pi± ui +
Z
ΓT
piT ui ,
(4.2.6)
∀u ∈ H 1D (ω ).
where H 1D (ω ) is the space of vector valued functions whose components and first derivatives are square integrable on ω , and whose value vanish on ΓD . For any given body
force density q = q i g i with q i in the dual space of H 1D (ω ) and traction surface force
densities p± and pT with contravariant components pi± and piT together defining a
1/2
functional on H 00 (Γ± ∪ ΓT ), this variational problem uniquely determine the displacement vector field v ∗ = vi∗ ∈ H 1D (ω ). With the unique displacement solution v ∗ of
the 3D elasticity equations determined, through the geometric equation (4.2.1) and the
constitutive equation (4.2.2), we can determine the stress tensor σ ∗ = σ ∗ij .
109
A stress field σ = σ ij is said to be statically admissible, if it satisfies the equilibrium equation (4.2.3) and the traction boundary condition (4.2.4). A displacement
field v = vi ∈ H 1 (Ω) is kinematically admissible, if it satisfies the clamping boundary
condition (4.2.5). If both σ and v are admissible, the following identity holds:
Z
Ω
Aijkl (σ kl − σ ∗kl )(σ ij − σ ∗ij ) +
Z
Z
=
Ω
Ω
C ijkl [χkl (v) − χkl (v ∗ )][χij (v) − χij (v ∗ )]
[σ ij − C ijkl χkl (v)][Aijkl σ kl − χij (v)]. (4.2.7)
This is the two energies principle. For spherical shells, our model derivation and justification are based on this identity.
4.3
Rescaled components
Due to the complicated expression (4.1.1), it is quite difficult to compute the row
divergence of a stress tensor given by its contravariant components. We will need to
verify the admissibility of a stress field in the justification of the spherical shell model,
and need to compute the residual of the equilibrium equation of a stress field for the
justification of the general shell model. So the calculation of the row divergence of stress
field is absolutely necessary. To simplify the calculation, we introduce the rescaled stress
components σ̃ ij for a stress field σ ij by defining
γβ
σ̃ αβ = ρµα
γσ ,
σ̃ 3α = ρσ 3α ,
σ̃ α3 = ρσ α3 ,
σ̃ 33 = ρσ 33 ,
(4.3.1)
110
or equivalently
σ αβ =
1 α γβ
ζ σ̃ ,
ρ γ
σ 3α =
1 3α
σ̃ ,
ρ
σ α3 =
1 α3
σ̃ ,
ρ
σ 33 =
1 33
σ̃ .
ρ
(4.3.2)
The following lemma indicates that, in terms of the rescaled stress components, the
divergence of the stress tensor σ ij has a simpler form.
Lemma 4.3.1. In terms of the rescaled components σ̃ ij , the row divergence of the stress
tensor σ ij has the expression
σ αj kj =
1 α γβ
γ
γ
ζ [σ̃ |β + µλ ∂t σ̃ λ3 − 2bτ σ̃ τ 3 ],
ρ γ
(4.3.3)
1
σ 3j kj = [σ̃ 3β |β + ∂t σ̃ 33 + bγλ σ̃ γλ ].
ρ
Note that, the derivatives in the right hand side are all taken with respect to the metric
of the middle surface of the shell.
Proof. For the first equation, on one hand, by the relations (4.1.6), we have
∗γ
∗γ
∗γ
ij
αj
αλ + Γ σ α3 + Γ∗α σ δτ + 2Γ∗α σ 3τ .
σ αj kj = ∂j σ αj + Γiγ σ αi + Γ∗α
3τ
ij σ = ∂j σ + Γλγ σ
δτ
3γ
On the other hand, we have
1 α
γ
γ
ζγ [ σ̃ γβ |β + µλ ∂t σ̃ λ3 − 2bτ σ̃ τ 3 ]
ρ
=
1 α
γ
γ
γ
ζ [(ρµτ σ τ β )|β + µλ ∂t (ρσ λ3 ) − 2bτ ρσ τ 3 ]
ρ γ
=
1 α
γ
γ
γ
γ
γ
ζ [∂ ρµ σ τ β + ρµ σ τ β + ρµτ σ τ β |β + µλ ∂t ρσ λ3 + ρ∂t σ λ3 − 2bτ ρσ τ 3 ]
τ |β
ρ γ β τ
111
=
1
1
β3
∂β ρσ αβ + ζλα µλ
σ τ β + σ αβ |β + ∂t ρσ α3 + ∂t σ α3 − 2ζλα bλ
βσ
τ
|β
ρ
ρ
∗γ
γ
α
γβ + ∂ σ αβ
= (Γγβ − Γγβ )σ αβ + (Γ∗α
β
γβ − Γγβ )σ
∗γ
β
λβ + Γ σ α3 + ∂ σ α3 + 2Γ∗α σ β3
+ Γγβ σ αγ + Γα
t
β3
λβ σ
γ3
∗γ
∗γ
γβ + Γ σ α3 + 2Γ∗α σ β3 .
= ∂β σ αβ + ∂t σ α3 + Γγβ σ αβ + Γ∗α
β3
γβ σ
γ3
The first equation in (4.3.3) then follows. In the above calculation, the following identities were used [30].
√
∂j g
∗i
Γij = √ ,
g
√
√
∂β g ∂β a
1
∗γ
γ
∂β ρ = √ − √ = Γγβ − Γγβ ,
ρ
g
a
√
∂t g
1
∗γ
∂t ρ = √ = Γγ3 .
ρ
g
For the second equation, we have
αβ + Γ∗λ σ 3n ,
σ 3j kj = ∂j σ 3j + Γ∗3
λn
αβ σ
and
1
[ σ̃ 3β |β + ∂t σ̃ 33 + bγλ σ̃ γλ ]
ρ
=
1
γ
[(ρσ 3β )|β + ∂t (ρσ 33 ) + bγλ ρµτ σ τ λ ]
ρ
∂β ρ 3β
∂ρ
γ
σ + ∂t σ 33 + t σ 33 + bγλ µτ σ τ λ
= σ 3β |β +
ρ
ρ
β
∗γ
γ
∗γ
τλ
= ∂β σ 3β + Γγβ σ 3γ + (Γγβ − Γγβ )σ 3β + ∂t σ 33 + Γγ3 σ 33 + Γ∗3
τ λσ
αβ + Γ∗λ σ 3n .
= ∂j σ 3j + Γ∗3
αβ σ
λn
The desired equation follows.
112
Note that except for some special shells, for example, plates and spherical shells,
the rescaled stress components σ̃ ij is not symmetric, more specifically, σ̃ 12 =
6 σ̃ 21 .
For consistency with the rescaled stress components, we introduce rescaled components for the applied forces. For the upper and lower surface force densities p± , we
introduce the rescaled components p̃i± by the relation
1
p± = pi± g i = p̃i± g i ,
ρ
(4.3.4)
where pi± are the usual contravariant components of the surface forces. The rescaled
components p̃i± take the diffences of the areas of the upper and lower surfaces from that
of the middle surface into account.
For the lateral surface force density pT , we introduce the rescaled components
p̃iT by the relation
1
pT = piT g i = p̃iT ai .
η
(4.3.5)
The rescaled components account the transverse area variation of the lateral surface,
and more explicitly express the dependence of the lateral surface force density on t.
For the body force density q, we define the new components q̃ i by
1
q = q i g i = q̃ i ai ,
ρ
(4.3.6)
where q i are the contravariant components of the body force density, while q̃ i are the
components of the body force density weighted by the transverse volume change, and
expressed in terms of the covariant basis on the middle surface.
113
In terms of the rescaled stress components and applied forces components, the
surface force condition (4.2.4) can be equivalently written as
j
σ̃ 3j = p̃+ on Γ+ ;
j
σ̃ 3j = −p̃− on Γ− ;
j
σ̃ jα nα = p̃T on ΓT .
(4.3.7)
The equilibrium residual σ ij kj + q i can be equally written as
σ αj kj + q α =
1 α γβ
γ
γ
ζγ [σ̃ |β + µλ ∂t σ̃ λ3 − 2bτ σ̃ τ 3 + q̃ γ ],
ρ
(4.3.8)
σ 3j kj + q 3 =
1 3β
[σ̃ |β + ∂t σ̃ 33 + bγλ σ̃ γλ + q̃ 3 ].
ρ
For the displacement vector v = vi g i , we introduce the rescaled components ṽi by
expressing the vector in terms of the basis vectors on the middle surface, i.e., v = ṽi ai .
In components, the relation is
γ
vα (x1 , x2 , t) = µα ṽγ (x1 , x2 , t),
v3 (x1 , x2 , t) = ṽ3 (x1 , x2 , t).
Lemma 4.3.2. In terms of the rescaled components ṽi of the displacement vector field v,
the strain tensor engendered by v can be expressed as
1 γ
1
γ
χαβ (v) = (ṽα|β + ṽβ|α − 2bαβ ṽ3 ) − t(bα ṽγ|β + bβ ṽγ|α − 2cαβ ṽ3 ),
2
2
1
γ
γ
χα3 (v) = χ3α (v) = (∂α ṽ3 + ∂t ṽα + bα ṽγ − tbα ∂t ṽγ ),
2
χ33 (v) = ∂t ṽ3 .
114
Proof. We need to compute the covariant derivatives vikj = ∂j v · g i . By direct
computation, we see
∂β v = ∂β ṽγ aγ + ṽγ ∂β aγ + ∂β ṽ3 a3 + ṽ3 ∂β a3 .
Using the definitions of the Christoffel symbols, curvature tensors, and covariant derivatives on the middle surface to the right hand sides of this equation, we get
γ
∂β v = (ṽγ|β − bγβ ṽ3 )aγ + (∂β ṽ3 + bβ ṽγ )a3 .
Therefore,
γ
λ
vαkβ = ∂β v · µλ
α aλ = µα (ṽλ|β − bλβ ṽ3 ) = ṽα|β − bαβ ṽ3 − tbα ṽγ|β + tcαβ ṽ3 ,
γ
v3kβ = ∂β v · g 3 = ∂β ṽ3 + bβ ṽγ .
γ
It is easy to see that ∂3 v = ∂t ṽγ aγ + ∂t ṽ3 a3 , so, we have vβk3 = ∂3 v · g β = µβ ∂t ṽγ =
γ
∂t ṽβ − tbβ ∂t ṽγ and v3k3 = ∂t ṽ3 . The lemma then follows from the definition of the
strain tensor (4.2.1).
115
Chapter 5
Spherical shell model
5.1
Introduction
In this chapter, we discuss the 2D modeling of the deformation of a thin shell
whose middle surface is a portion of a sphere. The shell can be totally or partially
clamped. The model is constructed in the vein of the minimum complementary energy
principle, and will be justified by the two energies principle. The form of the model is
similar to that of the plane strain cylindrical shells justified in Chapter 2, and can be put
in the abstract framework of Chapter 3. Since the membrane–shear operator B does not
have closed range, the behavior of the model solution is more complicated, and the justification is more difficult. For totally clamped spherical shells, convergence in the relative
energy norm of the 2D model solution to the 3D elasticity solution is proved. A convergence rate of O(1/6 ) in the relative energy norm is established under some smoothness
assumption on the shell data in the usual Sobolev sense. For partially clamped spherical
shells, convergence and convergence rate will be proved under a condition imposed on an
-independent 2D problem. This condition is an indirect requirement on the regularity
of the shell data, whose interpretation in the usual Sobolev sense is not completely clear
yet. An example for which the shell model might not be applicable will be given.
116
The spherical shell problem is another example that can be resolved by the two
energies principle. Together with the plane strain cylindrical shells, these special shell
problems provide examples for all kinds of shells as classified in the next chapter.
5.2
Three-dimensional spherical shells
A spherical shell is a special shell, to which all the definitions and equations of
Chapter 4 apply. Here, we summarize the things that are special to spherical shells.
The middle surface S of the spherical shell is a portion of a sphere of radius R. A
spherical shell, with middle surface S and thickness 2 , is a 3D elastic body occupying
the domain Ω ⊂ R3 , which is the image of a plate-like domain ω through the mapping
Φ defined in Chapter 4. We assume < R so that the mapping Φ is injective. Through
the mapping Φ, the Cartesian coordinates on ω furnish the curvilinear coordinates on
the shell Ω . The peculiarity of the spherical shell Ω lies in the fact that the mixed
α
curvature tensor of its middle surface is a scalar multiple of the Kronecker δ: bα
β = bδβ ,
with b = −1/R. To see this, we introduce the spherical coordinates on the middle
surface (x1 for the longitudes and x2 for the latitudes) and let the normal direction
point outward. With these coordinates, the covariant components of metric tensor are
a11 = R2 cos2 x2 , a22 = R2 , a12 = a21 = 0.
The covariant components of the curvature tensor are
b11 = −R cos2 x2 , b22 = −R, b12 = b21 = 0.
117
α
Note that, the mixed curvature tensor is given by bα
β = bδβ . We know that when
the curvilinear coordinates are changed, the mixed components of a second order tensor
change according to the rule of similarity matrix transformation. Therefore, on a sphere,
the mixed curvature tensor always takes this special form, no matter what coordinates
are used. Because of the special form of the mixed curvature tensor, we have the following
special relations that will substantially simplify the analysis.
α
µα
β = (1 − bt)δβ ,
H = b,
K = b2 ,
ζβα =
1
δα.
1 − bt β
ρ = (1 − bt)2 ,
η = 1 − bt,
(5.2.1)
g α = (1 − bt)aα ,
gαβ = (1 − bt)2 aαβ ,
gα =
1
aα ,
1 − bt
bαβ = baαβ ,
cαβ = b2 aαβ .
For the spherical shell, the rescaled stress components that was defined for general
shells in (4.3.1) become
σ̃ αβ = (1 − bt)3 σ γβ ,
σ̃ α3 = σ̃ 3α = (1 − bt)2 σ 3α ,
σ̃ 33 = (1 − bt)2 σ 33 .
(5.2.2)
Note that the matrix of rescaled stress components is symmetric, a property particular
to spherical shells. By using the equation (4.3.3), we can write the divergence of a stress
118
field in terms of the rescaled stress components as
σ αj ||j =
1
[σ̃ αβ |β + (1 − bt)∂t σ̃ α3 − 2bσ̃ α3 ],
(1 − bt)3
(5.2.3)
1
[σ̃ 3β |β + ∂t σ̃ 33 + baγλ σ̃ γλ ].
σ 3j ||j =
(1 − bt)2
The shell is subjected to surface forces p± on Γ± , and pT on ΓT per unit area. It
is loaded by a body force q per unit volume. The shell is clamped on ΓD . The rescaled
components of the applied forces are connected to the contravariant components through
the relations, see (4.3.4), (4.3.5), and (4.3.6),
1
p± = pi± g i = p̃i± g i ,
ρ
1
pT = piT g i = p̃iT ai ,
η
1
q = q i g i = q̃ i ai .
ρ
(5.2.4)
In terms of the rescaled stress components σ̃ ij and the rescaled applied force
components, the equilibrium equation σ ij kj + q i = 0 can be equivalently written as, see
(4.3.8),
σ̃ αβ |β + (1 − bt)∂t σ̃ 3α − 2bσ̃ 3α + q̃ α = 0,
(5.2.5)
σ̃ 3β |β + ∂t σ̃ 33 + baγλ σ̃ γλ + q̃ 3 = 0.
The unit outer normal vector on the upper surface Γ+ is obviously given by n+ = g 3
j
and on the lower surface Γ− , n− = −g 3 . The surface force conditions σ ij ni = p± on Γ±
are equivalent to
σ̃ 3α () = p̃α
+,
σ̃ 3α (− ) = −p̃α
−,
σ̃ 33 () = p̃3+ ,
σ̃ 33 (− ) = −p̃3− .
(5.2.6)
119
On the lateral surface ΓT , let the unit outer normal vector at a point on the middle
curve γT be n = nα aα , which should be in the middle surface. Note that along the
vertical straight fiber through this point, the unit outer normal should not change, so
n∗ = n∗i g i = nα aα . The components are n∗α = (1 − bt)nα ,
n∗3 = 0. The lateral surface
j
force condition σ ij n∗i = pT on ΓT , can be equivalently written as
σ̃ αβ nβ = p̃α
T,
σ̃ 3β nβ = p̃3T .
(5.2.7)
In terms of the rescaled applied surface force components, we define the odd and weighted
even parts of the surface forces by
α
p̃α
+ − p̃−
α
po =
,
2
α
p̃α
+ + p̃−
α
pe =
,
2
p̃3+ − p̃3−
3
po =
,
2
p̃3+ + p̃3−
3
pe =
.
2
(5.2.8)
For the body force, we define the components of the transverse average by
Z 1
i
q · ai dt.
qa =
2 −
We assume that the body force density q is constant in the transverse coordinate. This is
equivalent to q = qai ai . Under this assumption, the rescaled body force components are
quadratic polynomials in t, and we have q̃ i = q0i + tq1i + t2 q2i , with q0i = qai , q1i = −2bqai ,
and q2i = b2 qai .
120
For the lateral surface force, we define the components of the transverse average
and moment by
Z 1
i
p · ai dt,
pa =
2 − T
pim =
Z 3
tp · ai dt.
2 3 − T
We assume that the lateral surface force density pT changes linearly in the transverse
coordinate, or equivalently pT = (pia + tpim )ai .
Under this assumption, the rescaled
lateral surface force components p̃iT are quadratic functions in t, and we have p̃iT =
pi0 + tpi1 + t2 pi2 ,
with pi0 = pia , pi1 = pim − bpia , and pi2 = −bpim . The following
analyses can be carried through if q̃ i and p̃iT are arbitrary quadratic polynomials in t.
The restriction on the body force density and lateral surface force density can be further
relaxed, see Remark 6.3.1.
5.3
The spherical shell model
The model is a 2D variational problem defined on the space H = H 1D (ω) ×
∼
1 (ω). The solution of the model is composed of five two variable functions
H 1D (ω) × HD
∼
that can approximately describe the shell displacement arising in response to the applied
loads and boundary conditions. For ( θ , u , w) ∈ H, we define
∼ ∼
1
γαβ (u , w) = (uα|β + uβ|α ) − baαβ w,
∼
2
1
ραβ ( θ ) = (θα|β + θβ|α ), τβ ( θ , u , w) = θβ + ∂β w + buβ ,
∼
∼ ∼
2
(5.3.1)
121
which give the membrane, flexural, and transverse shear strains engendered by the displacement functions ( θ , u , w). The model reads: Find ( θ , u , w ) ∈ H, such that
∼ ∼
∼ ∼
Z
√
1 2
aαβλγ ρλγ ( θ )ραβ (φ ) adx
∼
∼
∼
3
ω
Z
Z
√
√
5
αβλγ
a
γλγ (u , w )γαβ (v , z) adx + µ
aαβ τβ ( θ , u , w )τα (φ , v , z) adx
+
∼
∼
∼ ∼
∼ ∼
∼ 6
∼
ω
ω
= hf 0 + 2 f 1 , (φ , y , z)i ∀ (φ , y , z) ∈ H, (5.3.2)
∼ ∼
∼ ∼
where aαβλγ = 2µaαλ aβγ + λ? aαβ aλγ is the 2D elasticity tensor of the shell and
λ? =
2µλ
.
2µ + λ
The leading term in the resultant loading functional is given by
Z
Z
√
√
λ
5
α
po τα (φ , y , z) adx −
p3o aαβ γαβ (y , z) adx
hf 0 , (φ , y , z)i =
∼
∼
∼
∼
∼
∼ ∼
6 ω
2µ + λ ω
Z
Z
√
α
α
α
3
α
3
+ [(qa − 2bpo + pe )yα + (qa + po |α + pe )z] adx +
pα
a yα , (5.3.3)
ω
∼
γT
and the higher order term is
Z
√
λ
hf 1 , (φ , y , z)i = −
(p3e + bp3o )aαβ ραβ (φ ) adx
∼ ∼
∼
∼
3(2µ + λ) ω
Z
√
1
+ b [bqaα yα + bqa3 z − (3pα
+ 2qaα )φα ] adx
e
∼
3 ω
Z
1
3
α
α
+
[−bpα
m yα + 2bpm z + (pm − bpa )φα ]. (5.3.4)
3 γT
122
Remark 5.3.1. It is noteworthy that the leading term p3a of the transverse component
of the lateral surface force is not incorporated in the expression of the leading term of
the resultant loading functional f 0 . Our explanation for this unreasonable phenomena is
that the effect of p3a is represented by the odd part of the upper and lower surface forces
pα
o through the compatibility condition (5.4.9).
This is the variational formulation of our spherical shell model. This model is a
close variant of the classical Naghdi model. The differences lie in the shear correction
factor 5/6, and more significantly, the expression of the flexural strain
1
ραβ = (θα|β + θβ|α ).
2
The flexural strain in the Naghdi model is given by
1
ρN
, w)
αβ = 2 (θα|β + θβ|α ) − bγαβ (u
∼
where γαβ is the membrane strain defined in (5.3.1).
We will derive a model for general shells in Chapter 6. When the general shell
model is applied to spherical shells, a spherical shell model that is slightly different from
the one we derived here will be obtained. Especially, the flexural strain will be given by
1
ραβ = (θα|β + θβ|α ) + bγαβ (u , w).
∼
2
It seems that the model (5.3.2) is closer to that of Budianski–Sanders [14].
123
We will prove the convergence of the spherical shell model in the next section, and
prove the convergence for general shell model in the next chapter. The discrepancy can
be explained by the difference in the resultant loading functional. What we can learn
from the difference between the two spherical shell models we derived is that the model
can be changed, but the crux is that the resultant loading functional must be changed
accordingly, otherwise, a variant in the model might lead to divergence.
To prove the well posedness of the classical Naghdi shell model, the following
equivalence was established in [11].
kρ N ( θ , u , w)kLsym (ω) + kγ (u , w)kLsym (ω) + kτ ( θ , u , w)kL (ω)
∼ ∼ ∼
∼ ∼
∼ ∼ ∼
2
∼2
∼
∼2
' k θ kH 1 (ω) + ku kH 1 (ω) + kwkH 1 (ω) ∀ ( θ , u , w) ∈ H,
∼ ∼
∼ ∼
∼ ∼
from which, by the observation
kρ ( θ )kLsym (ω) + (1 + |b|)kγ (u , w)kLsym (ω) + kτ ( θ , u , w)kL (ω)
∼ ∼
∼ ∼
∼ ∼ ∼
2
∼2
∼
∼2
& kρ N ( θ , u , w)kLsym (ω) + kγ (u , w)kLsym (ω) + kτ ( θ , u , w)kL (ω) ,
∼ ∼ ∼
∼ ∼
∼ ∼ ∼
2
∼2
∼
∼2
the following equivalency easily follows:
kρ ( θ )kLsym (ω) + kγ (u , w)kLsym (ω) + kτ ( θ , u , w)kL (ω)
∼ ∼
∼ ∼
∼ ∼ ∼
2
∼2
∼
∼2
' k θ kH 1 (ω) + ku kH 1 (ω) + kwkH 1 (ω) ∀ ( θ , u , w) ∈ H. (5.3.5)
∼ ∼
∼ ∼
∼ ∼
124
Since the elastic tensor aαβλγ and the contravariant metric tensor of the middle
surface aαβ are uniformly positive definite and bounded, the bilinear form in the left
hand side of the variational equation (5.3.2) is continuous and uniformly elliptic over the
space H. Therefore, we have
Theorem 5.3.1. If the resultant loading functionals (5.3.3) and (5.3.4) are linear con1 (ω), the model (5.3.2) has
tinuous functionals on the space H = H 1D (ω) × H 1D (ω) × HD
∼
∼
a unique solution ( θ , u , w ) in this space.
∼ ∼
Remark 5.3.2. The condition on the loading functionals for the existence of the model
solution can be met, if, say, the loading functions satisfy the conditions
p̃3± ∈ L2 (ω),
qai ∈ L2 (ω),
p̃α
± ∈ H (div, ω),
∼
pia , pim ∈ H −1/2 (∂T ω).
(5.3.6)
For simplicity, the flexural, membrane, and shear strains engendered by the model
solution will be denoted by
ραβ = ραβ ( θ ),
∼
5.4
= γ (u , w ),
γαβ
αβ ∼
τα = τα ( θ , u , w ).
∼ ∼
Reconstruction of the admissible stress and displacement fields
From the model solution ( θ , u , w ), we can reconstruct a statically admissible
∼ ∼
stress field and a kinematically admissible displacement field by explicitly giving their
components, and compute the constitutive residual so that we can use the two energies
125
principle to bound the error of the model solution in the energy norm. We will see that
the constitutive residual is formally small. A rigorous justification, which crucially hinges
on the asymptotic behavior of the model solution, will be given in the next section.
5.4.1
The admissible stress and displacement fields
Based on the model solution ( θ , u , w ) ∈ H, we define the following 2D tensors
∼ ∼
αβ
αβ
σ0 , σ1 , and a 2D vector σ03α by
αβ
σ0
αβ
σ1
+
= aαβλγ γλγ
= aαβλγ ρλγ +
λ
p3 aαβ ,
2µ + λ o
λ
(p3 + bp3o )aαβ ,
2µ + λ e
(5.4.1)
5
σ03α = [µaαβ τβ − pα
o ].
4
By using the model equation, it is readily checked that, in weak sense, these tensor- and
vector-valued functions satisfy the following system of differential equations.
1 2 αβ
2
2 2 α
σ1 |β − σ03α = 2 bpα
e + 3 b qa ,
3
3
2
1
αβ
σ0 |β − bσ03α = 2bpα
− pα
− (1 + b2 2 )qaα ,
o
e
3
3
(5.4.2)
2
1 2 3
αβ
3
baαβ σ0 + σ03α |α = −pα
o |α − pe − (1 + 3 b )qa
3
and the boundary conditions on γT
1 2 α
αβ
αβ
α
α
3α
2 3
σ 0 n β = pα
a − 3 b pm , σ1 nβ = pm − bpa , σ0 nα = bpm .
(5.4.3)
126
Indeed, if we substitute (5.4.1) into the above equations and boundary conditions, and
write the resulting equations and boundary conditions in weak form, we will get the
model equation (5.3.2). This is in fact how the model was derived.
αβ
αβ
and σ03α furnish the principal part of the statically ad-
The functions σ0 , σ1
missible stress field. To complete the construction of the stress field, we need to define
αβ
another 2D tensor-valued function σ2
αβ
The tensor-valued function σ2
and two scalar valued functions σ033 and σ133 .
will be determined by the equation
αβ
σ2 |β = −4bσ03α − b2 2 qaα
(5.4.4)
and the boundary condition
αβ
σ2 nβ = − 2 bpα
m on γT .
(5.4.5)
αβ
This equation and boundary condition together do not uniquely determine σ2 . We will
choose one so that
αβ
kσ2 kL (ω) . |b|kσ03α kL (ω) + b2 2 kqaα kL (ω) + |b| 2 kpα
m kH −1/2 (∂ ω)
2
2
2
T
∼
∼
∼
(5.4.6)
∼
holds. This is possible in view of Theorem 6.3.1 below.
The other two scalar functions σ033 and σ133 are explicitly defined by
σ033 =
σ133 =
1 2
αβ
3
(baαβ σ1 + pα
e |α − 2bqa ),
2
1
2
αβ
αβ
3
2 2 3
[baαβ σ0 + baαβ σ2 + pα
o |α + pe + (1 + b )qa ].
2
3
(5.4.7)
127
With all these tensor-, vector-, and scalar-valued 2D functions determined, we
define the rescaled components σ̃ ij of the stress field by
αβ
αβ
αβ
σ̃ αβ = σ0 + tσ1 + r(t)σ2 ,
α
3α
σ̃ 3α = σ̃ α3 = pα
o + tpe + q(t)σ0 ,
(5.4.8)
σ̃ 33 = p3o + tp3e + q(t)σ033 + s(t)σ133 ,
where r(t), q(t), and s(t) were defined in (2.4.6). From the definition, it is obvious that
the surface force conditions (5.2.6) on the upper and lower surfaces are precisely satisfied.
By using the boundary conditions (5.4.3), (5.4.5), and the compatibility condition
3
2 3
pα
o nα = pa − bpm ,
3
3
pα
e nα = pm − bpa
on γT ,
(5.4.9)
we can verify that the lateral surface force condition (5.2.7) is also exactly satisfied.
By using the equation (5.4.2), (5.4.4), and the definition (5.4.7), after a straightforward calculation, we can verify that the equilibrium equation (5.2.5) is precisely satisfied by the constructed rescaled stress components. Therefore, the functions σ̃ ij defined
by (5.4.8) are the rescaled components of a statically admissible stress field σ, whose
128
contravariant components, by the relation (5.2.2), are given by
σ αβ =
1
αβ
αβ
αβ
[σ0 + tσ1 + r(t)σ2 ],
3
(1 − bt)
σ 3α = σ α3 =
σ 33 =
1
α
3α
[pα
o + tpe + q(t)σ0 ],
(1 − bt)2
(5.4.10)
1
[p3o + tp3e + q(t)σ033 + s(t)σ133 ].
(1 − bt)2
Remark 5.4.1. On the shell edges Γ+ ∩ΓT and Γ− ∩ΓT , where the upper and lower surfaces meet the lateral surface, the surface forces exerted on the upper and lower surfaces
must be compatible with the force applied on the lateral surface in the sense that
p+ · n∗ () = pT · g 3 on Γ+ ∩ ΓT ,
p− · n∗ (− ) = −pT · g 3 on Γ− ∩ ΓT .
This compatibility condition is precisely equivalent to (5.4.9).
The kinematically admissible displacement field v is defined by giving its rescaled
components as
, ṽ = w + tw ,
ṽα = uα + tθα
3
1
(5.4.11)
in which w1 ∈ HD (ω) is a correction function to the transverse deflection whose definition
will be given in the next section. The clamping boundary condition on ΓD is obviously
satisfied.
129
5.4.2
The constitutive residual
For the admissible stress field σ and displacement field v constructed in the pre-
vious subsection, we denote the residual of the constitutive equation by %ij = Aijkl σ kl −
χij (v). By Lemma 4.3.2, the covariant components of strain tensor χij (v) engendered
by the displacement v defined in (5.4.11) are
+ tρ ) − bt(1 − bt)w a ,
χαβ (v) = (1 − bt)(γαβ
1 αβ
αβ
(5.4.12)
1
1
χ3α (v) = χα3 (v) = τα + t∂α w1 ,
2
2
χ33 (v) = w1 .
For the admissible stress field σ defined by (5.4.10), we can compute Aijkl σ kl by
using the definition of the 3D compliance tensor, the relations (5.2.1), and the definition
(5.4.1). The results are
+ tρ ) − bt2
Aαβkl σ kl = (1 − bt)(γαβ
αβ
−
λ
(p3 + bp3o )aαβ
2µ(2µ + 3λ) e
λ
[q(t)σ033 + s(t)σ133 ]aαβ
2µ(2µ + 3λ)
+ (1 − bt)r(t)
1
λ
λγ
[aαλ aβγ −
aαβ aλγ ]σ2 ,
2µ
2µ + 3λ
(5.4.13)
A3αkl σ kl =
1
β
β
3β
aαβ [po + tpe + q(t)σ0 ],
2µ
A33kl σ kl =
1
2(µ + λ) 3
{
[po + tp3e + q(t)σ033 + s(t)σ133 ]
2
2µ(1 − bt) 2µ + 3λ
130
−
λ
αβ
αβ
αβ
(1 − bt)[σ0 + tσ1 + r(t)σ2 ]aαβ }.
2µ + 3λ
Subtracting (5.4.12) from (5.4.13), we get the explicit expression of the residual
%ij :
%αβ = −bt2
−
λ
(p3 + bp3o )aαβ
2µ(2µ + 3λ) e
λ
[q(t)σ033 + s(t)σ133 ]aαβ
2µ(2µ + 3λ)
+ (1 − bt)r(t)
1
λ
λγ
[a a −
a a ]σ
2µ αλ βγ 2µ + 3λ αβ λγ 2
+ b(1 − bt)tw1 aαβ ,
%3α =
1
4
1
1
3β
β
[q(t) − ]aαβ σ0 − t aαβ pe − t∂α w1 ,
2µ
5
2µ
2
%33 =
1
1
λ
)−w
(
p3o −
aαβ γαβ
1
2
2µ + λ
(1 − bt) 2µ + λ
+
bt
λ
αβ
aαβ σ0
2
2µ(2µ
+
3λ)
(1 − bt)
+
2(µ + λ) 3
1
[
(tpe + q(t)σ033 + s(t)σ133 )
2
2µ(1 − bt) 2µ + 3λ
−
λ
αβ
αβ
(1 − bt)(tσ1 + r(t)σ2 )aαβ ].
2µ + 3λ
In the next section, we will prove that under some assumptions,
5
σ03α = [µaαβ τβ − pα
o] → 0
4
(5.4.14)
131
αβ
as → 0. By the estimate (5.4.6), we know that σ2
will converge to zero. From the
definition (5.4.7) we know that σ033 and σ133 are formally small. To make %33 small, we
1 (ω) to minimize
will choose w1 ∈ HD
[
λ
1
p3o −
aαβ γαβ (u , w )] − w1 .
∼
2µ + λ
2µ + λ
(5.4.15)
At the same time, due to the involvement of t∂α w1 in the expression of %3α , the quantity kw1 kH 1 (ω) needs to be small. With all these considerations, we can expect the
constitutive residual to be small.
5.5
Justification
The formal observations we made in the last subsection do not furnish a rigorous
justification, since the applied forces and the model solution may depend on the shell
thickness in an unexpected way. To prove the convergence, we need to make some
assumptions on the applied loads, and have a good grasp of the behavior of the model
solution when the shell thickness tends to zero. When → 0, everything may tend to
zero, so to prove the convergence, we need to consider the relative error. In addition to
the upper bound that can be obtained by bounding the constitutive residual, we need
to have a lower bound on the model solution.
5.5.1
Assumption on the applied forces
Henceforth, we assume that all the applied force functions explicitly involved in
the resultant loading functional in the model are independent of . I.e., we assume that
132
the functions
pio , pie , qai , pia , pim are independent of .
(5.5.1)
This assumption is different from the usual assumption adopted in asymptotic
theories, according to which, the functions −1 pio , rather than pio themselves, should
have been assumed to be independent of . Our assumption on pie and qai is the same as
the usual assumption [18].
Our assumption will reveal the potential advantages of using the Naghdi-type
model over the Koiter-type model. The convergence theorem can also be proved under
the usual assumption on the applied forces, but in that case, the difference between the
two types of models is negligible.
5.5.2
Asymptotic behavior of the model solution
Under the loading assumption (5.5.1), the shell model (5.3.2) fits into the abstract
-dependent variational problem (3.2.2) of Chapter 3. To apply the abstract theory, we
1 (ω)
define the following spaces and operators. As above H = H 1D (ω) × H 1D (ω) × HD
∼
∼
sym
with the usual product norm. We let U = L2
∼
(ω) with the equivalent inner product
√
1
(ρ 1 , ρ 2 )U =
aαβλγ ρ1λγ ρ2αβ adx ∀ ρ 1 , ρ 2 ∈ U,
∼ ∼
∼
∼ ∼
3 ω
Z
and define A : H → U , the flexural strain operator, by
A( θ , u , w) = ρ ( θ ) ∀ ( θ , u , w) ∈ H.
∼ ∼
∼ ∼
∼ ∼
133
sym
We also define B : H → L2
∼
(ω) × L2 (ω), combining the membrane and shear strain
∼
operators, by
B( θ , u , w) = [γ (u , w), τ ( θ , u , w)] ∀ ( θ , u , w) ∈ H.
∼ ∼
∼ ∼
∼ ∼ ∼
∼ ∼
The equivalence (5.3.5) guaranteed the condition (3.2.1) required by the abstract theory.
A totally or partially clamped spherical shell is stiff in the sense that it does not
allow for non-stretching deformations. If γ (u , w) = 0, we must have u = 0 and w = 0
∼ ∼
∼
[18]. Therefore, ker B = 0. According to the classification of the abstract -dependent
variational problem in Section 3.5, a spherical shell can never be a flexural shell.
For spherical shells, the most significant difference from the plane strain cylindrical
shells is that the operator B does not have closed range. We need to consider the space
sym
(ω) × L2 (ω), in which the norm is defined by
∼
∼
W = B(H) ⊂ L2
k[γ (u , w), τ ( θ , u , w)]kW = k( θ , u , w)kH .
∼ ∼
∼ ∼ ∼
∼ ∼
Equipped with this norm, W is a Hilbert space isomorphic to H. The operator B is, of
course, surjective from H to W .
sym
The space V is defined as the closure of W in L2
∼
(ω) × L2 (ω), with the inner
∼
product
√
1 γ 2 √adx + 5 µ
((γ 1 , τ 1 ), (γ 2 , τ 2 ))V =
aαβλγ γλγ
aαβ τβ1 τα2 adx ,
αβ
∼ ∼
∼ ∼
∼
∼
6 ω
ω
Z
Z
sym
which is equivalent to the inner product of L2
∼
(ω) × L2 (ω).
∼
134
The range of the operator B then is dense in V , as was required by the abstract
theory. The space V actually is equal to the product of V0 , the closure of the range of
sym
membrane strain operator γ in L2
∼
∼
(ω), and the closure of the range of shear strain
operator τ in L2 (ω). The latter, since the range of τ is dense in L2 (ω), is just equal to
∼
∼
∼
∼
L (ω). Therefore, we have the factorization
∼2
V = V0 × L2 (ω).
∼
(5.5.2)
According to the discussions in Section 3.4, the leading resultant loading functional f 0 determines the asymptotic behavior of the model solution.
Since ker B = 0 and B is an onto mapping from H to W , by the closed range
theorem, there exists a ζ∗0 ∈ W ∗ , such that the leading term in the loading functional
can be equivalently written as
hf 0 , (φ , y , z)i = hζ∗0 , B(φ , y , z)i ∀ (φ , y , z) ∈ H.
∼ ∼
∼ ∼
∼ ∼
We recall that without further assumption, the solution of this essentially singular perturbation problem is untractable. To sort out the tractable situations, we imposed the
condition (3.3.24) on ζ∗0 in Chapter 3. Namely,
ζ∗0 ∈ V ∗ .
(5.5.3)
135
This condition is equivalent to the requirement that the loading functional can be written
as
hf 0 , (φ , y , z)i = hζ∗0 , B(φ , y , z)iV ∗ ×V = (ζ 0 , B(φ , y , z))V ,
∼ ∼
∼ ∼
∼ ∼
here ζ 0 ∈ V is the Riesz representation of ζ∗0 ∈ V ∗ . Therefore the condition (5.5.3) is
equivalently requiring the existence of (γ 0 , τ 0 ) ∈ V = V0 × L2 (ω) such that
∼
∼ ∼
Z
hf 0 , (φ , y , z)i =
∼ ∼
ω
0 γ (y , z)√adx + 5 µ
aαβλγ γλγ
αβ ∼
∼ 6
Z
ω
√
aαβ τβ0 τα (φ , y , z) adx . (5.5.4)
∼ ∼
∼
Recalling the expression of the leading loading functional
Z
Z
√
√
λ
5
α
hf 0 , (φ , y , z)i =
po τα (φ , y , z) adx −
p3o aαβ γαβ (y , z) adx
∼
∼
∼ ∼
∼
∼
∼
6 ω
2µ + λ ω
Z
Z
√
α
α
α
3
α
3
pα
+ [(qa − 2bpo + pe )yα + (qa + po |α + pe )z] adx +
a yα , (5.5.5)
∼
ω
γT
we can see that the condition (5.5.4) is equivalent to the existence of κ ∈ V0 , such that
∼
Z
ω
√
aαβλγ κλγ γαβ (y , z) adx
∼
Z
=
ω
∼
α
3
α
3 √
[(qaα − 2bpα
+
o + pe )yα + (qa + po |α + pe )z] adx
∼
Z
γT
pα
a yα
1 (ω). (5.5.6)
∀ (y , z) ∈ H 1D (ω) × HD
∼
∼
136
Note that the second term in (5.5.5) can be equally written as
−
Z
Z
√
√
λ
λ
p3o aαβ γαβ (y , z) adx = −
aαβλγ aλγ p3o γαβ (y , z) adx ,
∼
∼
∼
∼
2µ + λ ω
2µ(2µ + 3λ) ω
(5.5.7)
so if p3o ∈ L2 (ω), we can determine γ 0 ∈ V0 as
∼
λ
0 =κ
3
γαβ
αβ − 2µ(2µ + 3λ) PV0 (aαβ po ),
sym
where PV0 is the orthogonal projection from L2
∼
(5.5.8)
(ω) to its closed subspace V0 , with
respect to the inner product of U .
By defining
τα0 =
1
β
a p ,
µ αβ o
(5.5.9)
we obtain ζ 0 = (γ 0 , τ 0 ) ∈ V such that the loading functional can be reformulated as
∼
∼
hf 0 , (φ , y , z)iH ∗ ×H = ((γ 0 , τ 0 ), B(φ , y , z))V
∼ ∼
∼
Z
=
ω
∼
∼ ∼
0 γ (y , z)√adx + 5 µ
aαβλγ γλγ
αβ ∼
∼ 6
Z
ω
√
aαβ τβ0 τα (φ , y , z) adx . (5.5.10)
∼ ∼
∼
Therefore, to use the abstract theory, the crux is the existence of κ ∈ V0 , such
∼
that the problem (5.5.6) is solvable. We will see that for totally clamped spherical shells,
under the data assumption (5.3.6), the existence of κ ∈ V0 is guaranteed automatically.
∼
But for partially clamped spherical shells, this requirement imposes a stringent restriction
on the applied forces. Even if the shell data are infinitely smooth, this requirement might
not be met.
137
Under the condition (5.5.6), the asymptotic behavior of the model solution follows
from Theorem 3.3.5 and (3.4.7). We have the convergence
lim [ kρ kLsym (ω) + kγ − γ 0 kLsym (ω) + kτ − τ 0 kL (ω) ] = 0.
∼ ∼2
∼
∼ ∼2
∼
∼ ∼2
→0
∼
(5.5.11)
∼
From this, we get the estimates
kρ kLsym (ω) . o(−1 ),
∼
kγ kLsym (ω) . 1,
∼
∼2
kτ kL (ω) . 1.
∼
2
∼
∼2
From the equivalency (5.3.5), we get the a priori estimates on the model solution
k θ kH 1 (ω) + ku kH 1 (ω) . o(−1 ),
∼
∼
∼
∼
(5.5.12)
kw kH 1 (ω) . k θ kL (ω) + ku kL (ω) .
∼
∼
2
2
∼
∼
If we assume more regularity on (γ 0 , τ 0 ), say,
∼
∼
(γ 0 , τ 0 ) ∈ [W, V ]1−θ,q ,
∼
∼
(5.5.13)
for some θ ∈ (0, 1) and q ∈ [1, ∞], or θ ∈ [0, 1] and q ∈ (1, ∞), according to Theorem 3.3.4
and (3.4.6), we have the stronger estimate on the asymptotic behavior of the model
solution:
kρ kLsym (ω) + kγ − γ 0 kLsym (ω) + kτ − τ 0 kL (ω) . K(, (γ 0 , τ 0 ), [V, W ]) . θ .
∼ ∼2
∼
∼ ∼2
∼
∼ ∼2
∼ ∼
∼
∼
(5.5.14)
138
And the estimates
kρ kLsym (ω) . θ−1 ,
∼
∼2
kγ kLsym (ω) . 1,
∼
kτ kL (ω) . 1.
∼
2
∼
∼2
By the equivalency (5.3.5), we get the a priori estimates
k θ kH 1 (ω) + ku kH 1 (ω) . θ−1 ,
∼
∼
∼
∼
(5.5.15)
kw kH 1 (ω) . k θ kL (ω) + ku kL (ω) .
∼
∼
2
2
∼
∼
The correction function w1 , based on its involvements in the constitutive residual,
will be defined as the solution of the variational equation
1
λ
0 , v)
2 (∇w1 , ∇v)L (ω) + (w1 , v)L (ω) = (
p3o −
aαβ γαβ
L2 (ω) ,
2
2
2µ + λ
2µ + λ
∼
(5.5.16)
1 (ω), ∀ v ∈ H 1 (ω).
w1 ∈ HD
D
Note that this definition of the correction function is not a simple analogue of the
definition of w1 in the plane strain cylindrical shell problems. Here we use γ 0 , rather
∼
than γ to define the correction. Due to the possible boundary layer of γ , if we put
∼
∼
γ in the place of γ 0 in (5.5.16), the convergence rate of the model solution will be
∼
∼
substantially reduced. Our correction on the transverse deflection is not an a posteriori
correction.
139
sym
From the definition (5.5.8) of γ 0 , we know γ 0 ∈ L2
∼
∼
∼
0 ∈
(ω), so we have aαβ γαβ
L2 (ω). By (3.3.38) in Theorem 3.3.6, we have
kw1 kH 1 (ω) + k − w1 −
λ
1
0 +
aαβ γαβ
p3 k
→ 0 ( → 0).
2µ + λ
2µ + λ o L2 (ω)
(5.5.17)
If we assume
0 − p3 ∈ [H 1 (ω), L (ω)]
λaαβ γαβ
2
o
1−θ,p
D
(5.5.18)
for some θ ∈ (0, 1) and p ∈ [1, ∞], or θ ∈ [0, 1] and p ∈ (1, ∞), by (3.3.36) in Theorem 3.3.6, we have
kw1 kH 1 (ω) + k − w1 −
λ
1
0 +
aαβ γαβ
p3 k
2µ + λ
2µ + λ o L2 (ω)
0 − p3 , [L (ω), H 1 (ω)]) . θ . (5.5.19)
. K(, λaαβ γαβ
2
o
D
5.5.3
Convergence theorem
With the estimates on the asymptotic behavior of the model solution established
in the previous subsection, and the expression of the constitutive residual (5.4.14), we
are ready to prove the convergence theorem. We denote the energy norms of a stress
field σ and a strain field χ by on the shell Ω by
Z
1
kσkE = (
Aijkl σ kl σ ij ) 2
Ω
Z
1
and kχkE = (
C ijkl χkl χij ) 2 ,
Ω
140
respectively. Since the elastic tensor C ijkl and the compliance tensor Aijkl are uniformly
positive definite and bounded, the energy norms are equivalent to the sums of the L2 (ω )
norms of the tensor components.
Theorem 5.5.1. Let v ∗ and σ ∗ be the displacement and stress fields on the spherical
shell arising in response to the applied forces and boundary conditions determined from
the 3D elasticity equations. And let v be the kinematically admissible displacement field
defined by (5.4.11) based on the model the solution ( θ , u , w ) and the correction func∼ ∼
tions w1 defined in (5.5.16), σ the statically admissible stress field defined by (5.4.10).
If there exists a κ ∈ V0 such that the functional reformulation (5.5.6) holds, then
∼
we have the convergence
kσ ∗ − σkE + kχ(v ∗ ) − χ(v)kE = 0.
kχ(v)kE →0
lim
(5.5.20)
If we further have the regularity (5.5.13) and (5.5.18), we have the convergence rate
kσ ∗ − σkE + kχ(v ∗ ) − χ(v)kE . θ .
kχ(v)kE (5.5.21)
Proof. We give the proof of (5.5.21). The proof of (5.5.20) is similar. For brevity, the
norm k · kL (ω ) will be denoted by k · k. Any function defined on ω will be viewed as a
2
function, constant in t, defined on ω .
141
First, we establish the lower bound for kχ(v)k2E . By the convergence (5.5.14),
we have
kρ kLsym (ω) . θ ,
∼
∼2
kγ − γ 0 kLsym (ω) . θ ,
∼
∼
∼2
kτ − τ 0 kL (ω) . θ .
∼
∼ ∼2
(5.5.22)
Since γ 0 and τ 0 can not be zero at the same time (otherwise f 0 = 0), we have
∼
∼
kγ kLsym (ω) + kτ kL (ω) ' kγ 0 kLsym (ω) + kτ 0 kL (ω) ' 1.
∼ ∼2
∼ ∼2
∼ ∼2
∼ ∼2
∼
(5.5.23)
∼
By the equivalence (5.3.5), we have k( θ , u , w )kH 1 (ω)×H 1 (ω)×H 1 (ω) . θ . The
∼ ∼
∼
∼
convergence (5.5.17) shows that
0 − p3 k
kw1 kH 1 (ω) . θ and kw1 kL (ω) ' kλaαβ γαβ
o L2 (ω) .
2
With all these estimates, it is easy to see that in the expression (5.4.12) of χij (v),
and τ dominate in χ
the terms γαβ
α
αβ and χ3α respectively, therefore,
2
X
α,β=1
kχαβ (v)k2 +
2
X
α=1
kχ3α (v)k2 & (kγ k2 sym
∼
L2
∼
(ω)
+ kτ k2L (ω) ) & ,
∼
2
∼
so,
kχ(v)k2E & .
(5.5.24)
142
From the two energies principle, we have
kσ ∗ − σk2E + kχ(v ∗ ) − χ(v)k2E =
Z
Ω
Aijkl %kl %ij .
3
X
k%ij k2 .
(5.5.25)
i,j=1
From the definition (5.4.1) and the definition of τβ0 , we have
αβ
σ0
αβ
σ1
+
= aαβλγ γλγ
= aαβλγ ρλγ +
λ
p3 aαβ ,
2µ + λ o
λ
(p3 + bp3o )aαβ ,
2µ + λ e
5
σ03α = [µaαβ (τβ − τβ0 )],
4
and so, by (5.5.22), we have the estimates
αβ
2 kσ1 k2 . 1+2θ ,
αβ
kσ0 k2 . ,
kσ03α k2 . 1+2θ .
(5.5.26)
αβ
By the estimate (5.4.6), we get kσ2 k2 . 1+2θ . From the last two equations of (5.4.7),
we see kσ033 k2 . 3+2θ ,
kσ133 k2 . 3 . Apply all the above estimations to the expression
of %αβ , it is readily seen that the square integral over ω of every term is bounded by
O(3 ), except the one in the third line, whose square integral on ω is bounded by
O(1+2θ ). Therefore we have k%αβ k2 . 1+2θ .
From the convergence (5.5.19), we know kw1 kH 1 (ω) . θ , so kt∂α w1 k2 . 1+2θ ,
together with (5.5.26), we have k%3α k2 . 1+2θ .
143
Our last concern is about %33 . In its expression, we equally write the first line as
1
1
λ
]−w
[
p3o −
aαβ γαβ
1
2
2µ + λ
(1 − bt) 2µ + λ
=
1
1
λ
3−
0 −w )
(
p
aαβ γαβ
1
o
2µ + λ
(1 − bt)2 2µ + λ
−
2 2
λ
1
αβ (γ − γ 0 ) + 2bt − b t w .
a
1
αβ
αβ
(1 − bt)2 2µ + λ
(1 − bt)2
By the convergence (5.5.22) and (5.5.19), wee see that the square integral of this expression is bounded by O(1+2θ ). The second and third lines are obviously bounded by
O(3 ). The last line is bounded by O(1+2θ ), so, we have k%33 k2 . 1+2θ . We obtained
the upper bound kσ ∗ − σk2E + kχ(v ∗ ) − χ(v)k2E . 1+2θ .
The estimate (5.5.21) follows from the lower bound (5.5.24) and this upper bound.
The proof of (5.5.20) is a verbatim repetition, except replacing 1+2θ by o(), and θ by
o(1).
Remark 5.5.1. If the odd part of the applied tangential surface forces pα
o is not zero,
the deformation violates the Kirchhoff–Love hypothesis. Actually, the estimate (5.5.22)
shows that the transverse shear strain converges to the finite limit
1
β
aαβ po .
µ
144
5.5.4
About the condition of the convergence theorem
The convergence theorem hinges on the existence of κ ∈ V0 , such that
∼
√
aαβλγ κλγ γαβ (y , z) adx
Z
∼
ω
Z
=
ω
∼
α
3
α
3 √
[(qaα − 2bpα
+
o + pe )yα + (qa + po |α + pe )z] adx
∼
Z
γT
pα
a yα
1 (ω). (5.5.27)
∀ (y , z) ∈ H 1D (ω) × HD
∼
∼
The membrane strain operator γ (y , z) defines a linear continuous operator γ : H 1D (ω) ×
∼ ∼
∼
∼
1 (ω) −→ V , whose range is dense in V . Since ker γ = 0, the function kγ (y , z)k sym
HD
0
0
L
∼
∼ ∼
∼2
1 (ω), which is weaker than the original norm.
defines a norm on the space H 1D (ω) × HD
∼
1 (ω) with respect to
In the notation of [18], we denote the completion of H 1D (ω) × HD
∼
]
]
this new norm by VM (ω). Obviously, γ can be extended uniquely to VM (ω), and the
∼
extended linear continuous operator, still denoted by γ , defines an isomorphism between
∼
]
]
VM (ω) and V0 . By the closed range theorem, for any f ∈ [VM (ω)]∗ , there exists a unique
κ ∈ V0 , such that
∼
√
]
aαβλγ κλγ γαβ (y , z) adx = hf, (y , z)i ]
∀ (y , z) ∈ VM (ω).
]
∗
∼
∼
∼
∼
[VM (ω)] ×[VM (ω)]
ω
Z
(5.5.28)
Therefore, the question of existence of κ ∈ V0 in (5.5.27) is equivalent to the
∼
1 (ω)
question that whether or not the linear functional defined on the space H 1D (ω) × HD
∼
by the right hand side of (5.5.27) can be extended to a linear continuous functional on
]
VM (ω).
145
Under some smoothness assumption on the boundary γ of the middle surface, the
following Korn-type inequality was established in [23] and [19]: There exists a constant
C such that for any u ∈ H 10 (ω), w ∈ L2 (ω)
∼
∼
ku k2 1
+ kwk2L (ω) ≤ Ckγ (u , w)k2 sym .
∼ H (ω)
∼ ∼
L
(ω)
2
(5.5.29)
∼2
∼
Therefore, if the shell is totally clamped, by this inequality, it is easily seen that
]
VM (ω) = H 10 (ω) × L2 (ω).
∼
In this case, the mild condition (5.3.6) is enough to guarantee the existence of κ ∈ V0 ,
∼
and therefore the convergence (5.5.20).
]
If the shell is partially clamped, the space VM (ω) can be huge and its norm can be
very weak, so that the existence of κ can not be guaranteed even if the loading functions
∼
are in D(ω), the space of test functions of distribution, see [38].
As to the convergence rate, the regularity requirements (5.5.13) and (5.5.18) are
quite abstract. Except for the cases in which we purposely load the shell in such a way
that the conditions are satisfied, we have no idea about how to explain them for partially
clamped shells.
For totally clamped spherical shells, under the smoothness assumption on the
3
α
1
3
2
3
2
α
1
shell data: γ ∈ C 4 , pα
o ∈ H (ω), pe ∈ H (ω), po ∈ H (ω), pe ∈ H (ω), qa ∈ H (ω),
qa3 ∈ H 2 (ω), we can prove
K(, (γ 0 , τ 0 ), [V, W ]) . 1/6
∼
∼
146
and
0 − p3 , [L (ω), H 1 (ω)]) . 1/2 .
K(, λaαβ γαβ
2
o
0
Therefore, the regularity conditions (5.5.13) and (5.5.18) hold for θ = 1/6, and so the
convergence rate in (5.5.21) can be determined as 1/6 . If the odd part of the tangential
surface forces vanishes, or very small, the value of θ is 1/5. See Section 6.6.4.
5.5.5
A shell example for which the model might fail
The condition ζ∗0 ∈ V ∗ , or equivalently, the reformulation (5.5.10) of the leading
term of the resultant loading functional, is necessary for our justification of the spherical
shell model (5.3.2). As we have seen, this condition is almost trivially satisfied for a
totally clamped spherical shell, but it imposes a stringent restriction on the shell data if
the shell is partially clamped. We give an example, for which the the condition can not
be satisfied, and so the model can not be justified.
Consider a partially clamped spherical shell not subject to any body force (q =
0), or upper and lower surface forces (p± = 0), loaded by lateral surface force pT =
−2 M α ). The vector valued function M α is
−2 tM α aα (pia = 0, p3m = 0, and pα
m =
defined on ∂T ω, and independent of . To get the physical meaning, we can imagine
applying a pure bending moment of fixed magnitude on the traction lateral surface of
a sequence of spherical shells with thickness tending to zero. With this load applied on
the 3D shell, the resultant loading functional in the model will be
Z
1
hf 0 + 2 f 1 , (φ , y , z)i =
M α (φα − byα ).
∼ ∼
3 γT
147
The condition ζ∗0 ∈ V ∗ is equivalent to, see (5.5.10), the existence of (γ 0 , τ 0 ) ∈ V0 ×
∼ ∼
L (ω), such that
∼2
Z
Z
Z
√
√
1
5
αβλγ
0
α
a
γλγ γαβ (y , z) adx + µ
aαβ τβ0 τα (φ , y , z) adx
M (φα − byα ) =
∼
∼ ∼
∼ 6
∼
3 γT
ω
ω
∀ (φ , y , z) ∈ H.
∼ ∼
Recalling the definition (5.3.1) of the operators γαβ and τα , we can see that this is a
condition impossible to satisfy, therefore the model (5.3.2) can not be justified for this
specially loaded spherical shell. The limiting membrane shell model has no solution for
this problem. Our model gives a solution in the space H, but convergence in the relative
energy norm can not be proved.
148
Chapter 6
General shell theory
6.1
Introduction
In this chapter, we present and justify the 2D model for general 3D shells. The
form of the model is similar to that of the cylindrical and spherical shell models of
Chapters 2 and 5. The model is a close variant of the classical Naghdi shell model,
which can be fit into the abstract -dependent variational problem of Chapter 3, and it
can be accordingly classified as a flexural shell or a membrane–shear shell. By proving
convergence of the 2D model solution to the 3D solution in the relative energy norm,
the model is completely justified for flexural shells and totally clamped elliptic shells.
The latter are special membrane–shear shells. Convergence in the relative energy norm
is also proved for other membrane–shear shells under the assumption that the applied
forces are “admissible”. Convergence rates are established, which are related to the shell
data in an abstract notion.
For general shells, the main difficulty to overcome is that, unlike for the special
shells, we can not construct a statically admissible stress field from the model solution,
so the two energies principle can not be used to justify the model. As an alternative,
we will reconstruct a stress field that is almost admissible, which has small residuals in
the equilibrium equation and lateral traction boundary condition. We will establish an
integration identity (6.3.17) to incorporate the equilibrium residual and lateral traction
149
boundary condition residual. This identity is a substitute to the two energies principle
for the analysis of general shells.
The model is justified for flexural shells in Section 6.5. In this case, convergence
in the relative energy norm can be proved without any assumption. If solution of the
limiting flexural model, which is an -independent problem, is assumed to have some
regularity in the notion of interpolation spaces, convergence rate of the 2D model solution
toward the 3D shell solution can be established. The theory will be applied to the
plate bending problem, which is a special flexural shell problem, to reproduce the plate
bending theory. We can use the known results for this special problem to argue that the
convergence rate we determined for flexural shells is the best possible.
We justify the model for totally clamped elliptic shells in Section 6.6. The reason
of sorting out these special membrane–shear shells is that totally clamped elliptic shells
possess some special properties, especially the Korn-type inequality (6.6.2), so that we
can prove the convergence theorem without making any assumption. Convergence rate
will be determined if the solution of the limiting membrane shell model has some regularity. With some smoothness in the usual Sobolev sense assumed on the shell data, the
convergence rate O(1/6 ) in the relative energy norm will be determined.
Section 6.7 is devoted to the justification of the model for all the other membrane–
shear shells. In the general situation, there are more difficulties to overcome. The model
justification can only be obtained under some restrictions on the applied forces. There
are two sources for the new difficulties, one is rooted in the model, the other is due
to the residuals of equilibrium equation and lateral traction boundary condition of the
reconstructed almost admissible stress field. The first one is resolved by concreting the
150
condition (3.3.24) that we introduced in Chapter 3 on the abstract level. The second will
be resolved by adopting the condition of “admissible applied forces” proposed in [18].
Under these assumptions, the convergence of the model solution to the 3D solution in
the relative energy norm will be proved.
To address the potential superiority of our Naghdi-type model over the Koitertype model, we make an assumption on the loading functions, which is slightly different
from what usually assumed in asymptotic theories. We will see that under this loading
assumption, there is no significant difference between the two types of models for flexural
shells. But for membrane–shear shells, it is very likely that the Koiter-type model does
not converge while the Naghdi-type model does. In Chapter 7, we will show that under
the usual assumption on the applied forces, the difference between these two types of
models is not significant.
6.2
The shell model
The general 3D shell problem is what was described in Chapter 4. The shell Ω
is assumed to be clamped on a part of its lateral surface ΓD . It is subjected to surface
traction force on the remaining part ΓT of the lateral surface, whose density is pT . The
shell is subjected to surface forces on the upper and lower surfaces Γ± , whose densities
are p± , and loaded by a body force with density q.
In terms of the rescaled surface force components p̃i± , see (4.3.4), we define the
odd and weighted even parts of the surface forces by
α
p̃α
+ − p̃−
pα
=
,
o
2
α
p̃α
+ + p̃−
pα
=
,
e
2
p̃3 − p̃3−
p3o = +
,
2
p̃3 + p̃3−
p3e = +
.
2
(6.2.1)
151
For the body force, we define the components of the transverse average by
Z 1
i
q · ai dt.
qa =
2 −
(6.2.2)
We assume the body force density is constant in the transverse coordinate. This assumption is equivalent q = qai ai . Under this assumption, the rescaled components of
the body force density q̃ i = ρqai , see (4.3.6), are quadratic polynomials of t, and we have
q̃ i = q0i +tq1i +t2 q2i , with q0i = qai , q1i = −2Hqai , and q2i = Kqai . The following calculation
can be carried through if q̃ i are arbitrary quadratic polynomials of t.
We assume that the rescaled lateral surface force components, see (4.3.5), are
quadratic functions of t. I.e. p̃iT = pi0 + tpi1 + t2 pi2 , with pi0 , pi1 , and pi2 independent of
t.
The restriction on the body force density and lateral surface force density can be
relaxed.
The model is a 2D variational problem defined on the space H = H 1D (ω) ×
∼
1 (ω). The solution of the model is composed of five two-variable functions
H 1D (ω) × HD
∼
that can approximately describe the shell displacement arising in response to the applied
loads and boundary conditions. For ( θ , u , w) ∈ H, we define the following 2D tensors.
∼ ∼
1
γαβ (u , w) = (uα|β + uβ|α ) − bαβ w,
∼
2
1
1
ραβ ( θ , u , w) = (θα|β + θβ|α ) + (bλ
u
+ bλ
α uβλ ) − cαβ w,
∼ ∼
2
2 β α|λ
τβ ( θ , u , w) = bλ
β uλ + θβ + ∂β w,
∼ ∼
(6.2.3)
152
which give the membrane strain, flexural strain, and transverse shear strain engendered
by the displacement functions ( θ , u , w). The model reads: Find ( θ , u , w ) ∈ H, such
∼ ∼
∼ ∼
that
Z
√
1 2
aαβλγ ρλγ ( θ , u , w )ραβ (φ , y , z) adx
∼ ∼
∼ ∼
∼
3
ω
Z
Z
√
√
5
aαβλγ γλγ (u , w )γαβ (y , z) adx + µ
+
aαβ τβ ( θ , u , w )τα (φ , y , z) adx
∼
∼
∼ 6
∼ ∼
∼ ∼
∼
ω
ω
= hf 0 + 2 f 1 , (φ , y , z)i ∀ (φ , y , z) ∈ H, (6.2.4)
∼ ∼
∼ ∼
in which the forth order 2D contravariant tensor aαβλγ is the elastic tensor of the shell,
defined by aαβλγ = 2µaαλ aβγ + λ? aαβ aλγ .
The resultant loading functionals are given by
Z
Z
√
√
5
λ
pα
τ
(φ
,
y
,
z)
p3o aαβ γαβ (y , z) adx
adx
−
α
o
∼ ∼
∼ 2µ + λ ω
∼
∼
∼ ∼
6 ω
Z
Z
√
γ
α
α
α
α
3
3
+ [(pe + qa − 2bγ po )yα + (po |α + pe + qa )z] adx +
pα
0 yα , (6.2.5)
hf 0 , (φ , y , z)i =
∼
ω
γT
and
√
1
1
2
γ
[ Kqaα yα + Kqa3 z − (bα
pe + Hqaα )φα ] adx
γ
∼
3
3
ω 3
Z
hf 1 , (φ , y , z)i =
∼ ∼
−
Z
√
λ
(p3e + 2Hp3o )aαβ ραβ (φ , y , z) adx
∼
∼ ∼
3(2µ + λ) ω
Z
1
3
+
(pα φα + pα
2 yα − 2p2 z). (6.2.6)
3 γT 1
153
Note that the leading term p30 of the transverse component of the lateral surface force is
not incorporated in the leading term of the resultant loading functional f 0 . The reason
is the same as what we remarked for the spherical shell model, see Remark 5.3.1.
This model is a close variant of the classical Naghdi shell model. See [49], [10],
[18], [11], [6], [15], etc., where the latter was cited or derived in various ways. This model
is different from the generally accepted Naghdi model in three ways. First, the resultant
loading functional is more involved. The noticeably different form of the leading term
f 0 is a consequence of our loading assumption. The classical loading functional is the
leading term of the functional defined in Section 7.5. The higher order term 2 f 1 does
not affect the convergence and convergence rate theorems in the relative energy norm.
See Section 7.1. Second, the coefficient of the transverse shear term is 5/6 rather than
the usual value 1. The third, and most significant, difference is in the expression of the
flexural strain ραβ . Comparing our expression
1
1
u
+ bλ
ραβ ( θ , u , w) = (θα|β + θβ|α ) + (bλ
α uβ|λ ) − cαβ w
∼ ∼
2
2 β α|λ
with that of Naghdi’s
1
1
ρN
θ , u , w) = (θα|β + θβ|α ) − (bλ
u
+ bλ
α uλ|β ) + cαβ w,
αβ (∼
∼
2
2 β λ|α
we see the relationship
γ
λ
ραβ = ρN
αβ + bα γλβ + bβ γγα ,
where γαβ is the membrane strain defined in (6.2.3).
(6.2.7)
154
To establish the well posedness of the classical Naghdi model, the following equivalency was proved in [11].
kρ N ( θ , u , w)kL2 + kγ (u , w)kL2 + kτ ( θ , u , w)kL2
∼
∼ ∼
∼
∼ ∼
∼ ∼ ∼
∼
∼
' k θ kH 1 + ku kH 1 + kwkH 1 ∀ ( θ , u , w) ∈ H,
∼ ∼
∼ ∼
∼ ∼
from which, by using the relation (6.2.7) and the observation
kρ ( θ , u , w)kL2 + (1 + 2B)kγ (u , w)kL2 + kτ ( θ , u , w)kL2
∼ ∼ ∼
∼ ∼
∼
∼
∼ ∼ ∼
∼
& kρ N ( θ , u , w)kL2 + kγ (u , w)kL2 + kτ ( θ , u , w)kL2 ,
∼
∼ ∼
∼
∼ ∼
∼ ∼ ∼
∼
∼
where B is the maximum absolute value of the components of the mixed curvature tensor
bα
β over ω, the following equivalency easily follows
kρ ( θ , u , w)kL2 + kγ (u , w)kL2 + kτ ( θ , u , w)kL2
∼ ∼ ∼
∼
∼ ∼
∼
∼ ∼ ∼
∼
' k θ kH 1 + ku kH 1 + kwkH 1 ∀ ( θ , u , w) ∈ H. (6.2.8)
∼ ∼
∼ ∼
∼ ∼
Since the elastic tensor aαβλγ and the contravariant metric tensor aαβ are uniformly positive definite and bounded, so the bilinear form in the left hand side of the
model (6.2.4) is continuous and uniformly elliptic on the space H. Therefore, we have
155
Theorem 6.2.1. If the resultant loading functional f 0 + 2 f 1 in the model (6.2.4)
defines a linear continuous functional on the space H, then the model has a unique
solution ( θ , u , w ) ∈ H.
∼ ∼
Remark 6.2.1. The condition of the this existence theorem can be met, if, for example,
the applied force functions satisfy the condition
p̃3± ∈ L2 (ω), qai ∈ L2 (ω), pα
(div, ω), pi0 , pi1 , pi2 ∈ H −1/2 (∂T ω).
±∈H
∼
(6.2.9)
Henceforth, we will assume that the loading functions satisfy this condition.
For brevity, the membrane, flexural, and transverse shear strains engendered by
the model solution will be denoted by
= γ (u , w ),
γαβ
αβ ∼
6.3
ραβ = ραβ ( θ , u , w ),
∼ ∼
τα = τα ( θ , u , w ).
∼ ∼
Reconstruction of the stress field and displacement field
From the model solution ( θ , u , w ), we can reconstruct a stress field σ by giv∼ ∼
ing its contravariant components, and a displacement field v by giving its covariant
components. The displacement field is kinematically admissible, but the stress field is
not exactly statically admissible since the equilibrium equation and the lateral surface
force condition on ΓT can not be precisely satisfied. We will compute the equilibrium
156
residual, lateral force condition residual, and the constitutive residual between the constructed stress and displacement fields, and establish an identity to express the errors of
the reconstructed stress and displacement fields in terms of all these residuals so that a
rigorous proof of the model convergence can be obtained by bounding these residuals.
6.3.1
The stress and displacement fields
Based on the model solution, we define the following 2D symmetric tensor-valued
αβ
αβ
functions σ0 , σ1 , and a 2D vector-valued function σ03α .
αβ
σ1
αβ
σ0
= aαβλγ ρλγ +
λ
(p3 + 2Hp3o )aαβ ,
2µ + λ e
2
λ
αβ
+
= H 2 σ1 + aαβλγ γλγ
p3 aαβ ,
3
2µ + λ o
(6.3.1)
5
σ03α = (µaαβ τβ − pα
o ).
4
It can be verified that, in weak sense, these tensor- and vector-valued functions satisfy
the following system of differential equations and boundary condition.
2 αβ
2
2
γ
σ1 |β − σ03α = 2 (bα
pe + Hqaα ),
γ
3
3
3
αβ
(σ0 −
αβ
1 2 β αγ
2
1
γ
dγ σ1 )|β − bα
σ03λ = 2bα
po − pα
− (1 + K 2 )qaα ,
γ
e
λ
3
3
3
bαβ (σ0 −
αβ
(σ0 −
(6.3.2)
1
1 2 β αγ
2
dγ σ1 ) + σ03α |α = −pα
|α − p3e − (1 + K 2 )qa3 ,
o
3
3
3
1 2 β αγ
1
αβ
3β
α
2 3
dγ σ1 )nβ = pα
+ 2 pα
0
2 , σ1 nβ = p1 , σ0 nβ = − p2 on γT . (6.3.3)
3
3
157
Indeed, by substituting (6.3.1) into (6.3.2), we will recover the model (6.2.4) in differential
form. In fact, this is how the model was derived.
These functions furnish the principal part of the stress field. To complete the
construction of the stress field, we need to define another 2D symmetric tensor-valued
αβ
function σ2
αβ
and two scalars σ033 and σ133 . The tensor-valued function σ2
is required
to satisfy the following equation and boundary condition
αβ
β αγ
3γ
2 α
(σ2 − 2 dγ σ1 )|β = −4bα
γ σ0 − K qa in ω,
(6.3.4)
αβ
β αγ
(σ2 − 2 dγ σ1 )nβ = 2 pα
2 on γT .
αβ
αβ
This system does not uniquely determine σ2 . We will choose σ2
to minimize its
sym
L
(ω) norm, see the next subsection. The scalars are explicitly defined by
∼2
σ033 =
2
αβ
3
(b σ + pα
e |α − 2Hqa ),
2 αβ 1
(6.3.5)
1
2
αβ
β αγ
αβ
β αγ
σ133 = [bαβ ((σ0 − 2 dγ σ1 ) + bαβ (σ2 − 2 dγ σ1 )
2
3
3
3
2
3
+ pα
o |α + pe + (1 + K)qa ]. (6.3.6)
158
With all these 2D functions determined, we define the contravariant components σ ij of
our stress field σ by
β
λγ
λγ
λγ
σ αβ = ζλα ζγ [σ0 + tσ1 + r(t)σ2 ],
1 α
3α
[p + tpα
e + q(t)σ0 ],
ρ o
σ 3α = σ α3 =
σ 33 =
(6.3.7)
1 3
[p + tp3e + q(t)σ033 + s(t)σ133 ],
ρ o
here
1
t2
r(t) = 2 − , q(t) = 1 −
3
t2
t
, s(t) = (1 −
2
t2
).
2
The stress field σ ij is obviously symmetric. Following classical terminologies, we call
αβ
σ0
αβ
the membrane stress resultant, σ1
αβ
the first membrane stress moment, and σ2
the second membrane stress moment.
By the definition (4.3.1), the rescaled components σ̃ ij of this stress field are
β
αγ
αγ
αγ
σ̃ αβ = ρζγ [σ0 + tσ1 + r(t)σ2 ],
α
3α
σ̃ 3α = σ̃ α3 = pα
o + tpe + q(t)σ0 ,
(6.3.8)
σ̃ 33 = p3o + tp3e + q(t)σ033 + s(t)σ133 .
β
β
β
By using the relation ρζγ = δγ − tdγ , the rescaled membrane stress components can be
written as
αβ
σ̃ αβ = (σ0 −
1 2 β αγ
αβ
αβ
β αγ
β αγ
αγ
dγ σ1 ) + tσ1 + r(t)(σ2 − 2 dγ σ1 ) − tdγ [σ0 + r(t)σ2 ].
3
159
By the definition (6.3.7), we easily see that the surface force conditions (4.2.4),
or equivalently, (4.3.7) on Γ± are precisely satisfied by the constructed stress field. To
simplify the verification of the lateral force condition, we write the rescaled lateral surface
force components as
1 2 α
α
α
2 α
α
α
2 α
p̃α
T = p0 + tp1 + t p2 = p0 + 3 p2 + tp1 + r(t) p2 ,
(6.3.9)
p̃3T = p30 + tp31 + t2 p32 = p30 + 2 p32 + tp31 − q(t) 2 p32 .
It can be verified that the compatibility condition of the applied surface forces on the
shell edges, see Remark 5.4.1, is equivalent to
3
2 3
α
3
pα
o n α = p 0 + p 2 , p e n α = p1 .
(6.3.10)
On the lateral boundary ΓT , by using the compatibility condition (6.3.10) and the boundary conditions imposed in (6.3.3) and (6.3.4), we get the residual of the lateral surface
force condition:
(σ αj − σ ∗αj )n∗j =
t β γλ
1 α γβ
γ
γλ
ζγ (σ̃ nβ − p̃T ) = − dλ [σ0 + r(t)σ2 ]nβ ζγα ,
η
η
(6.3.11)
(σ 3j − σ ∗3j )n∗j =
1 3α
(σ̃ nα − p̃3T ) = 0.
η
160
By using the identities (4.3.8), the equations in (6.3.2), and the equation in (6.3.4),
we can get the residual of the equilibrium equation:
t
β γλ
β γλ
σ αj kj + q α = − ζγα [dλ σ0 + r(t)dλ σ2 ]|β ,
ρ
(6.3.12)
t
β γλ
β γλ
σ 3j kj + q 3 = − bαβ [dλ σ0 + r(t)dλ σ2 ].
ρ
Formally, these residuals are small. More importantly, they are explicitly expressible in
terms of the two-variable functions, so they are not far beyond our control.
The displacement field v is defined by giving its rescaled components:
, ṽ = w + tw + t2 w ,
ṽα = uα + tθα
3
1
2
(6.3.13)
1 are two correction functions that will be defined later. This
in which w1 , w2 ∈ HD
correction does not affect the basic pattern of the shell deformation which has already
been well captured by the primary displacement functions ( θ , u , w ) given by the shell
∼ ∼
model. Obviously, v is kinematically admissible.
6.3.2
The second membrane stress moment
αβ
In the construction of the stress field, the tensor-valued function σ2
was required
to satisfy
αβ
β αγ
3γ
2 α
(σ2 − 2 dγ σ1 )|β = −4bα
γ σ0 − K qa in ω
(6.3.14)
αβ
β αγ
(σ2 − 2 dγ σ1 )nβ = 2 pα
2 on ∂T ω.
161
The weak form of this problem is
Z
dλ v
+ dλ
α vβ|λ √
αβ β α|λ
αβ vα|β + vβ|α √
2
σ1
σ2
adx = adx
∼
∼
2
2
ω
ω
Z
Z
+
ω
√
3λ
2
α
(4bα
+ 2
λ σ0 + Kqa )vα adx
∼
Z
γT
pα
v ∈ H 1D . (6.3.15)
2 vα ∀ ∼
∼
We have
Theorem 6.3.1. Among all symmetric tensor-valued functions satisfying the equation
(6.3.15), we can choose one such that
αβ
αβ
kσ2 kL2 . Bkσ03α kL2 + 2 Bkσ1 kL2 + 2 Kkqaα kL2 + 2 kpα
2 kH −1/2 (∂ ω) , (6.3.16)
∼
∼
∼
∼
T
∼
where B = max{|bα
β |}.
To prove this theorem, we need some lemmas. Let γ̄αβ be the linear continuous
sym
operator from H 1D to L2
∼
∼
defined by
γ̄αβ (v ) =
∼
vα|β + vβ|α
∀ v ∈ H 1D (ω).
∼ ∼
2
We have
Lemma 6.3.2. The operator γ̄αβ is injective.
Proof. The equation γ̄αβ (v ) = 0 is a system of three first order PDE’s:
∼
λ
∂1 v1 − Γλ
11 vλ = 0, ∂2 v2 − Γ22 vλ = 0,
∂1 v2 + ∂2 v1
− Γλ
12 vλ = 0,
2
162
and we have the boundary condition vλ = 0 on ∂D ω. The difference of the first two
equations and the third equation together with the boundary condition constitute an
elliptic system for the variables v1 and v2 , see [25], with the Cauchy data imposed
on part of the domain boundary. Therefore, by the unique continuation theorem of
Hörmander [31], v1 and v2 must be identically equal to zero.
Lemma 6.3.3. The operator γ̄αβ defines an isomorphism between H 1D and a closed sub∼
sym
space L̄2
∼
sym
of L2
∼
.
Proof. Considering the compact operator A2 : H 1D −→ (L2 )3 defined by
∼
λ
λ
A2 (v ) = (Γλ
11 vλ , Γ12 vλ , Γ22 vλ )
∼
and treating γ̄αβ as the operator A1 in Lemma 2.3.2, the following inequality then follows
from the Korn’s inequality of plane elasticity,
kv kH 1 . kγ̄αβ (v )kLsym .
∼
∼
∼2
∼D
Therefore, the operator γ̄αβ has closed range.
Proof of Theorem 6.3.1. We consider the three terms in the right hand side of
αβ
(6.3.15) separately, and write σ2
αβ
αβ
αβ
= σ2,1 + σ2,2 + σ2,3 . The estimate (6.3.16) will follow
αβ
from superposition. The tensor valued function σ2,1 is required to satisfy the equation
Z
dλ v
+ dλ
α vβ|λ √
αβ β α|λ
αβ vα|β + vβ|α √
2
adx = σ1
adx .
σ2,1
∼
∼
2
2
ω
ω
Z
163
sym
We define the linear continuous operator D1 : L̄2
∼
sym
by
∼
−→ L2
λ
dλ
vα|β + vβ|α
β vα|λ + dα vβ|λ
D1 (
)=
,
2
2
which is the composition of the inverse of γ̄αβ and a self explanatory operator. It is
αβ
αβ
obvious that the symmetric tensor σ2,1 = 2 D1∗ (σ1 ) satisfies the above equation. Here
D1∗ is the dual of D1 .
αβ
The tensor valued function σ2,2 is required to satisfy the equation
Z
ω
αβ vα|β + vβ|α √
σ2,2
2
Z
adx =
∼
ω
sym
We consider the operator D2 : L̄2
∼
3λ + 2 Kq α )v √adx ∀ v ∈ H 1 .
σ
(4bα
a α
0
λ
∼
∼
∼D
−→ L2 defined by
∼
D2 (
vα|β + vβ|α
2
)= v
∼
which is the composition of the inverse of γ̄αβ and the identical inclusion of H 1D in
∼
αβ
3λ
2
α
L . The symmetric tensor determined by σ2,2 = D2∗ (4bα
λ σ0 + Kqa ) satisfies this
∼2
equation.
αβ
The tensor valued function σ2,3 is required to satisfy the equation
Z
αβ vα|β + vβ|α √
2
σ2,3
adx = pα
v ∈ H 1D .
2 vα ∀ ∼
∼
∼
2
ω
γT
Z
164
sym
We consider the operator D3 : L̄2
∼
D3 (
1/2
−→ H 00 (∂T ω) defined by
∼
vα|β + vβ|α
)= v
∼
2
which is the composition of the inverse of γ̄αβ and the trace operator. The symmetric
αβ
tensor determined by σ2,3 = 2 D3∗ (pα
2 ) satisfies this equation. By superposition, the
solution of (6.3.15) can be chosen as
αβ
σ2
αβ
3λ
2
α
2 ∗ α
= 2 D1∗ (σ1 ) + D2∗ (4bα
λ σ0 + Kqa ) + D3 (p2 ).
The theorem then follows from the fact that D1∗ , D2∗ , and D3∗ are bounded operators
sym
from L2
∼
sym
, L2 , and H −1/2 (∂T ω) to L2
∼
∼
∼
sym
Note that L̄2
∼
, respectively.
sym
is a closed subspace of L2
∼
is in consistence with the nonuniqueness
of the solution of (6.3.14).
6.3.3
The integration identity
Due to the residuals of the traction boundary condition (6.3.11) and the equilib-
rium equation (6.3.12), we can not have the two energies principle (4.2.7) precisely hold
for the constructed stress and displacement fields. However, for these fields, we have the
identity (6.3.17) below, which is a substitute to the two energies principle.
165
Theorem 6.3.4. For the stress field σ and displacement field v defined by (6.3.7) and
(6.3.13), we have the following identity
Z
Ω
Aijkl (σ kl − σ ∗kl )(σ ij − σ ∗ij ) +
Z
=
Ω
Z
Ω
C ijkl [χkl (v) − χkl (v ∗ )][χij (v) − χij (v ∗ )]
[Aijkl σ kl − χij (v)][σ ij − C ijkl χkl (v)] + r, (6.3.17)
in which
Z Z r=2
ω −
√
γ
β
β
γ
t[dλ σ0αλ + r(t)dλ σ2αλ ](ζα vγ∗ − ζα vγ )|β adtdx
∼
Z Z −2
ω −
√
β γλ
β γλ
tbγβ [dλ σ0 + r(t)dλ σ2 ](v3∗ − v3 ) adtdx , (6.3.18)
∼
and v ∗ = vi∗ and σ ∗ = σ ∗ij are the displacement and stress fields on the shell determined
from the 3D elasticity equations respectively.
Proof. For a stress field σ = σ ij and an admissible displacement field v, the
following identity follows from an integration by parts over the shell Ω .
Z
Ω
Aijkl (σ kl − σ ∗kl )(σ ij − σ ∗ij ) +
Z
=
Z
−2[
Γ+
Ω
Z
Ω
C ijkl [χkl (v) − χkl (v ∗ )][χij (v) − χij (v ∗ )]
[σ ij − C ijkl χkl (v)][Aijkl σ kl − χij (v)] + 2
(σ 3i −σ ∗3i )(vi∗ −vi )−
Z
Γ−
Z
(σ 3i −σ ∗3i )(vi∗ −vi )+
Ω
(σ ij kj + q i )(vi∗ − vi )
Z
ΓT
(σ αi −σ ∗αi )n∗α (vi∗ −vi )].
(6.3.19)
166
Substituting the displacement field v (6.3.13), the stress field σ (6.3.7), the residual of the lateral surface force condition (6.3.11), and the residual of the equilibrium
equation (6.3.12) into the integration identity (6.3.19), we get
Z
Aijkl (σ kl − σ ∗kl )(σ ij − σ ∗ij ) +
Ω
Z
=
Ω
Z
Ω
C ijkl [χkl (v) − χkl (v ∗ )][χij (v) − χij (v ∗ )]
[σ ij − C ijkl χkl (v)][Aijkl σ kl − χij (v)] − 2
Z
Ω
t
β γλ
β γλ
b [d σ + r(t)dλ σ2 ](v3∗ − v3 )
ρ αβ λ 0
Z
−2
Z
t α β γλ
t β γλ
β γλ
γλ
∗ −v )+2
∗ −v ).
ζγ [dλ σ0 +r(t)dλ σ2 ]|β (vα
dλ [σ0 +r(t)σ2 ]nβ ζγα (vα
α
α
ρ
η
Ω
ΓT
(6.3.20)
The key observation here is that the last two integrals in this identity can be merged
into a single term. Recalling the meanings of ρ and η, we can convert the integrals to
the coordinate domain ω and write the sum of the last two integrals as
Z Z
−2
− ω
√
β
β
γ
t[dλ σ0αλ + r(t)dλ σ2αλ ]|β ζα (vγ∗ − vγ ) adx dt
∼
Z Z
+2
− ∂T ω
q
β
γ
tdλ [σ0αλ + r(t)σ2αλ ]nβ ζα (vγ∗ − vγ )
aαβ ẋα ẋβ dsdt.
Note that the covariant derivatives in this expression are all taken with respect to the
q
metric on the middle surface S and aαβ ẋα ẋβ ds = dγ is just the arc length element of
γT . For each t ∈ (− , ) we can use the Green’s theorem (4.1.4) on the middle surface
S. The above two-term sum is equal to
Z Z
2
− ω
√
β
β
γ
γ
t[dλ σ0αλ + r(t)dλ σ2αλ ](ζα vγ∗ − ζα vγ )|β adx dt.
∼
167
The second integral in the right hand side of the equation (6.3.20) can also be converted
to the coordinate domain ω . The desired identity then follows.
The above calculations are formal because the stress field σ might not be eligible
for integration by parts. Note the fact that the possible singularities of the stress field σ
arise at the lateral boundary points where the type of boundary condition changes. We
can get around this difficulty by approximating v ∗ − v by infinitely smooth functions
with compact supports in the domain (∂T ω ∪ ω) × [− , ]. Here, we assume that ∂T ω =
∂ω − ∂D ω.
Using the fact that r(t) is an even function of t and the expression (6.3.13), we
can further write the expression of r as
Z Z r=2
ω −
√
β
β
γ
t[dλ σ0αλ + r(t)dλ σ2αλ ](ζα vγ∗ )|β adtdx
∼
Z Z −2
ω −
β γλ
β γλ √
tbγβ [dλ σ0 + r(t)dλ σ2 ]v3∗ adtdx
Z Z −2
ω −
∼
√
β
β
t2 [dλ σ0αλ + r(t)dλ σ2αλ ]θα|β adtdx
∼
Z Z +2
ω −
√
β γλ
β γλ
t2 bγβ [dλ σ0 + r(t)dλ σ2 ]w1 adtdx . (6.3.21)
∼
The identity (6.3.17) expresses the energy norms of the errors of the stress field σ and
displacement field v in terms of the constitutive residual between these two fields, and
the extra term r. Note that in the expression of r, the 3D displacement v ∗ was involved.
To bound r, in addition to knowledge of the behavior of the model solution, some bounds
on the 3D displacement v ∗ will also be needed.
168
Remark 6.3.1. If the body force density q is not constant in the transverse coordinate,
the additional term
Z
2
ω
γ
∗ − v ) + (q̃ 3 − ρq 3 )(v ∗ − v )]
[ζγα (q̃ γ − ρqa )(vα
α
3
a 3
needs to be added to the expression (6.3.18) of r. Under the assumption
kq̃ i − ρqai kL (ω ) . o(3/2 ),
2
(6.3.22)
the ensuing analyses can be carried through, and the convergence theorems can be proved
in all the cases.
If the rescaled lateral surface force components q̃Ti are not quadratic polynomials in
t, we can replace q̃Tα by their quadratic Legendre expansions q̄Tα , and q̃T3 by the quadratic
interpolation q̄T3 at the points − , 0, , see (6.3.9) for explanations, and we need to add
yet another term
Z
Z −2
∂T ω − γ
∗
3
3
∗
α
[(p̄α
T − p̃T )ζα (vα − vα ) + (p̄T − p̃T )(v3 − v3 )]
to the expression of r. The following convergence theorems can be proved in all the cases
if
kp̄iT − p̃iT kL [∂ ω×(− ,)] . o(3/2 ).
2 T
(6.3.23)
169
6.3.4
Constitutive residual
In this subsection, we compute the constitutive residual %ij = Aijkl σ kl − χij (v)
for the stress field σ (6.3.7) and the displacement field v (6.3.13) constructed previously.
First, by using Lemma 4.3.2 and the definition (6.2.3), we can compute the strain tensor
engendered by the displacement v. The result is
+ tρ − t(bλ γ + bλ γ ) + (tc
2
χαβ (v) = γαβ
α λβ
αβ − bαβ )(tw1 + t w2 )
αβ
β λα
1
γ γ − t2 (bα θγ|β
+ bβ θγ|α
),
2
1
1
χα3 (v) = χ3α (v) = τα + (t∂α w1 + t2 ∂α w2 ),
2
2
χ33 (v) = w1 + 2tw2 . (6.3.24)
Next, by using the definition of the 3D compliance tensor Aijkl , the relation (4.1.5),
the formulae (6.3.7), the definition (6.3.1), and the identities
1
2λ
2(2µ + λ)
(a b aλγ + bαλ aβγ aλγ −
b a aλγ ) =
b
2µ αλ βγ
2µ + 3λ αβ λγ
2µ(2µ + 3λ) αβ
and
2λ
1
λ
(aαλ bβγ aλγδρ τδρ + bαλ aβγ aλγδρ τδρ −
b a aλγδρ τδρ ) = bλ
α τλβ + bβ τλα
2µ
2µ + 3λ αβ λγ
for any symmetric tensor ταβ , after a lengthy calculation, we get the following expressions
for Aijkl σ kl :
+ tρ − t(bλ γ + bλ γ )
Aαβkl σ kl = γαβ
α λβ
αβ
β λα
170
+
λ
[a (p3 + tp3e ) + 2t(Haαβ − bαβ )p3o ]
2µ(2µ + 3λ) αβ o
2
λ
+ H 2 2 [ραβ +
a (p3 + 2Hp3o )]
3
2µ(2µ + 3λ) αβ e
2
2λ
−(t2 + Ht 2 )[bλ
ρλβ + bλ
ρλα −
b (p3 + 2Hp3o )]
α
β
3
2µ(2µ + 3λ) αβ e
A3αkl σ kl =
+
t2
λ
λγ
λγ
λγ
(bαλ bβγ −
bαβ aλγ )[σ0 + tσ1 + r(t)σ2 ]
2µ
2µ + 3λ
+
λ
r(t)
λγ
λγ
[(aαλ − tbαλ )(aβγ − tbβγ )σ2 −
g a σ ]
2µ
2µ + 3λ αβ λγ 2
−
λ
g [p3 + tp3e + q(t)σ033 + s(t)σ133 ],
2µ(2µ + 3λ) αβ o
t
1
3γ
γ
γ
aαγ [po + σ0 q(t)] +
{aαγ pe
2µ
2µ
+ (tcαγ − 2bαγ +
A33kl σ kl =
(6.3.25)
2H − tK
3γ
γ
γ
gαγ )[po + tpe + q(t)σ0 ]},
ρ
1 3
1
ν
αβ
αβ
αβ
(po − νaαβ σ0 ) + t (p3e − νaαβ σ1 ) − r(t)aαβ σ2
E
E
E
1
+ {q(t)σ033 + s(t)σ133
E
2H − tK 3
+t
[po + tp3e + q(t)σ033 + s(t)σ133 ]},
ρ
(6.3.26)
(6.3.27)
where
E=
µ(2µ + 3λ)
λ
and ν =
µ+λ
2(µ + λ)
are the Young’s modulus and Poisson ratio of the elastic material comprising the shell.
Combining (6.3.1), (6.3.24), (6.3.25), (6.3.26), and (6.3.27), after some calculations, we get the explicit expression of the constitutive residual:
%αβ =
2 2 2 2
1
λ ρλα ) + t2 (bλ
H ραβ − (t2 + Ht 2 )(bλ
ρλβ + bλ
α
α θλ|β + bβ θλ|α )
β
3
3
2
171
+
λ
t2
λγ
λγ
λγ
(bαλ bβγ −
bαβ aλγ )[σ0 + tσ1 + r(t)σ2 ]
2µ
2µ + 3λ
−
λ
g [q(t)σ033 + s(t)σ133 ]
2µ(2µ + 3λ) αβ
+
r(t)
λ
λγ
λγ
[(aαλ − tbαλ )(aβγ − tbβγ )σ2 −
g a σ ]
2µ
2µ + 3λ αβ λγ 2
+
λ
4
2
[(2tH + H 3 2 )aαβ + 4H(t2 + Ht 2 )bαβ − t2 cαβ ]p3o
2µ(2µ + 3λ)
3
3
+
2
4
λ
[ H 2 2 aαβ + (4t2 + Ht 2 )bαβ − t3 cαβ ]p3e
2µ(2µ + 3λ) 3
3
+(bαβ − tcαβ )(tw1 + t2 w2 ),
%3α =
(6.3.28)
1
4
t
t2
aαλ σ03λ [q(t) − ] − ∂α w1 − ∂α w2
2µ
5
2
2
+
t
2H − tK
λ
3λ
{aαλ pλ
+ (tcαλ − bαλ +
gαλ )[pλ
e
o + tpe + q(t)σ0 ]}, (6.3.29)
2µ
ρ
1 3
1
αβ
αβ
(po − νaαβ σ0 ) − w1 ] + t[ (p3e − νaαβ σ1 ) − 2w2 ]
E
E
1
αβ
+ [q(t)σ033 + s(t)σ133 − νaαβ r(t)σ2 ]
E
t 2H − tK 3
[po + tp3e + q(t)σ033 + s(t)σ133 ].
+
E
ρ
%33 = [
(6.3.30)
Remark 6.3.2. If we had not defined the flexural strain ραβ different from that of
λ
λ
Naghdi’s (ρN
αβ ), there would be an additional term t(bα γλβ + bβ γλα ) in the residual
%αβ . Our variant does make the constitutive residual smaller, at least formally.
Based on their involvements in (6.3.29) and (6.3.30), the two correction functions
w1 and w2 will be chosen to make
1 3
λ
αβ
+ 1 p3 − ν 3 H 2 a σ αβ −w (6.3.31)
(po −νaαβ σ0 )−w1 = −
aαβ γαβ
1
αβ 1
E
2µ + λ
2µ + λ o E 2
172
and
1 3
αβ
(p − νaαβ σ1 ) − 2w2
E e
=−
λ
1
2Hλ2
aαβ ραβ +
p3e −
p3 − 2w2 (6.3.32)
2µ + λ
2µ + λ
µ(2µ + λ)(2µ + 3λ) o
small in the L2 (ω) norm. At the same time, their H 1 (ω) norms must be kept under
control.
These formulae make it possible to prove the the model convergence. A rigorous
justification of the model requires a great deal of information about the behavior of the
model solution, and we must consider the relative energy norm. In addition to the upper
bound on the residuals, we also need a lower bound on the energy contained in the 2D
model solution. Since the 3D solution v ∗ was involved in the extra term r in the identity
(6.3.17), we also need to bound the 3D solution. To this end, we need a Korn-type
inequality on thin shells.
6.3.5
A Korn-type inequality on three-dimensional thin shells
In this subsection, we establish an inequality to bound the term r in the inte-
gration identity (6.3.17). With this inequality, we will be able to show that the extra
term r, which is due to the the residuals of equilibrium equation and lateral surface force
condition of our almost admissible stress field, do not affect the convergence of the model
solution toward the 3D solution in the cases of flexural shells and totally clamped elliptic
shells. For all the other membrane–shear shells, this inequality will be used to prove the
convergence theorem under some other assumptions on the loading functions.
173
It is well known that Korn’s inequality, which bounds the H 1 norm of a displacement field by its strain energy norm, contains a constant depending on the shape and
size of the elastic body. On a thin shell, the H 1 norm of a displacement field can not be
bounded by its strain energy norm uniformly with respect to the shell thickness. The
following -dependent inequality (6.3.35) of Korn-type was established in [18] and [22],
see also [32] and [1] for similar results.
Let ω 1 = ω × (−1, 1) be the scaled coordinate domain, and ∂D ω 1 = ∂D ω × [−1, 1]
be the part of the scaled clamping lateral boundary. For any v ∈ H 1D (ω 1 ), we define a
displacement field v on Ω by
t
v (x , t) = v(x , ) ∀ x ∈ ω, t ∈ (− , ).
∼
∼ ∼
(6.3.33)
We define the scaled strain tensor for the vector field v by
χij (v) = χij (v ).
(6.3.34)
There exists an 0 > 0, such that when ≤ 0 the inequality
kvk2 1 1 . −2
H D (ω )
3
X
kχij (v)k2
i,j=1
uniformly holds for all and v ∈ H 1D (ω 1 ).
From this inequality, we immediately have
L2 (ω 1 )
(6.3.35)
174
Theorem 6.3.5. There exists a constant 0 > 0 such that, for all ≤ 0 and any
v = vi ∈ H 1D (ω ), we have
2
X
α=1
6.4
kvα k2 1
+ kv3 k2L (ω ) . −2
H (ω)×L2 (− ,)
2
3
X
i,j=1
kχij (v)k2L (ω ) .
2
Classification
The shell model (6.2.4) is an -dependent variational problem whose solution
can behave dramatically different in different circumstances. To get accurate a priori
estimates, the problem must be classified. By making some assumptions on the applied
forces, we can fit the shell model into the abstract problem (3.2.2) of Chapter 3, and
accordingly classify the problem.
6.4.1
Assumptions on the loading functions
We assume all the loading functions explicitly involved in the model, namely, the
odd and weighted even parts pio and pie of the applied surface forces, the coefficients pi0 ,
pi1 , and pi2 of the rescaled lateral surface force, and the components qai of the body force,
are independent of .
Roughly speaking, the convergence theory established under this assumption has
the physical meaning that when the model is applied to a realistic shell, no matter how
the shell is loaded, the thinner the shell the better the results the model provides.
175
6.4.2
Classification
To use the results of Chapter 3, we introduce the following spaces and operators.
1 (ω) with the usual product norm. We let U =
As above, H = H 1D (ω) × H 1D (ω) × HD
∼
∼
sym
L
(ω) with the equivalent inner product
∼2
(ρ 1 , ρ 2 )U =
∼
∼
Z
√
1
aαβλγ ρ1λγ ρ2αβ adx ∀ ρ 1 , ρ 2 ∈ U,
∼
∼ ∼
3 ω
and define A : H → U , the flexural strain operator, by
A( θ , u , w) = ρ ( θ , u , w) ∀ ( θ , u , w) ∈ H.
∼ ∼
∼ ∼ ∼
sym
We also define B : H → L2
∼
∼ ∼
(ω) × L2 (ω), combining the membrane and shear strain
∼
operators, by
B( θ , u , w) = [γ (u , w), τ ( θ , u , w)] ∀ ( θ , u , w) ∈ H.
∼ ∼
∼ ∼
∼ ∼ ∼
sym
We introduce the space W = B(H) ⊂ L2
∼
∼ ∼
(ω) × L2 (ω), in which the norm is defined
∼
by
k[γ (u , w), τ ( θ , u , w)]kW =
∼ ∼
∼ ∼ ∼
inf
[γ ( ū ,w̄),τ ( θ̄ , ū ,w̄)]=[γ (u ,w),τ ( θ ,u ,w)]
∼ ∼
∼ ∼∼
∼ ∼
k( θ̄ , ū , w̄)kH
∼ ∼
∼ ∼∼
∀ [γ (u , w), τ ( θ , u , w)] ∈ W.
∼ ∼
∼ ∼ ∼
Equipped with this norm, W is a Hilbert space isomorphic to H/ ker(B). The operator
B is, of course, an onto mapping from H to W .
176
sym
The space V is defined as the closure of W in L2
∼
(ω) × L2 (ω), with the inner
∼
product
√
1 γ 2 √adx + 5 µ
aαβλγ γλγ
aαβ τβ1 τα2 adx ,
((γ 1 , τ 1 ), (γ 2 , τ 2 ))V =
αβ
∼ ∼
∼ ∼
∼
∼
6 ω
ω
Z
Z
sym
which is equivalent to the inner product of L2
∼
(ω) × L2 (ω).
∼
The range of the operator B then is dense in V , as was required by the abstract
theory of Chapter 3. The space V actually is equal to the product of V0 , the closure of
sym
the range of γ in L2
∼
∼
(ω), and the closure of the range of τ in L2 (ω). The latter, since
∼
∼
the range of τ is dense in L2 (ω), is just equal to L2 (ω), so we have the factorization
∼
∼
∼
V = V0 × L2 (ω).
(6.4.1)
∼
From the definitions of the membrane, flexural, and shear strains (6.2.3), we
easily see that A and B are continuous operators. The equivalency (6.2.8) guaranteed
the condition (3.2.1).
Remark 6.4.1. It should be noted that, in contrast to the fact that V is a product space,
the space W can not be viewed as a product space generally. If the shell is flat, the membrane strain will be separated from the flexural and shear strains. The model is split to the
Reissner–Mindlin plate stretching model and bending model. When the flat shell (plate)
sym
is totally clamped, the space W can be identified as [a closed subspace of L2
∼
] × H̊ (rot),
∼
see [8], [13] and [35]. For the plane strain cylindrical shell problems, the operator B has
177
closed range, and so W is simply equal to V . For general shells, the characterization of
W in the Sobolev sense is either unclear or impossible.
Under the loading assumption, in the resultant loading functional f 0 + 2 f 1 of
the model (6.2.4), both f 0 and f 1 , see (6.2.5) and (6.2.6), are independent of , so the
loading functional is rightfully in the form of the right hand side of the abstract problem
(3.2.2) of Chapter 3. According to the classification of Section 3.5, if f 0 |ker B 6= 0, the
shell problem is called a flexural shell. If f 0 |ker B = 0, then, since B is surjective from
H to W , by the closed range theorem, there exists a unique ζ∗0 ∈ W ∗ such that
hf 0 , ( θ , u , w)i = hζ∗0 , B( θ , u , w)i ∀ ( θ , u , w) ∈ H.
∼ ∼
∼ ∼
∼ ∼
If ζ∗0 ∈ V ∗ , the shell problem is called a membrane–shear shell. If f 0 |ker B = 0, but ζ∗0
is not in V ∗ , the shell model is not justified.
The kernel space ker B, according to the definition of the operator B, is composed of admissible displacement fields of the form (uα + tθα )aα + wa3 , from which the
engendered membrane strain γ (u , w) and the transverse shear strain τ ( θ , u , w) vanish.
∼ ∼
∼ ∼ ∼
The displacement in this space is pure flexural. In this kind of deformation, the intrinsic
metric of the middle surface does not change infinitesimally, and there is no transverse
shear strain. The condition f 0 |ker B 6= 0 means that the applied forces do bring about
the pure flexural deformation. Thus the name flexural shell.
In the case of membrane–shear shells, the membrane energy and transverse shear
energy together dominate the strain energy. It seems that there is no way to distinguish
178
the contributions to the total energy from the membrane and the transverse shear strains.
This is the reason why we call the shells in the second category the membran–shear shells.
Flexural shells, of course, require ker B 6= 0 (pure flexural deformation is not
inhibited). Membrane–shear shells include two different kinds, namely, ker B = 0 (pure
flexural is inhibited, henceforth, shells of this kind will be called stiff shells) and ker B 6= 0
but f 0 |ker B = 0 (pure flexural is not inhibited, but the loading function does not make
the pure flexural happen). A typical example of this second kind membrane–shear shells
is plate stretching.
6.5
Flexural shells
We prove the convergence of the 2D model solution toward the 3D solution in the
relative energy norm for flexural shells as classified in the last section. The convergence
can be proved without any extra assumption. Under some regularity assumption on the
solution of the limiting flexural model (6.5.3), convergence rate (as a power of ) will be
established.
First, we resolve the term r in the identity (6.3.17). From (6.3.18), we see that
|r| . [kσ 0 kLsym (ω ) + kσ 2 kLsym (ω ) ][kv ∗ − v kH 1 (ω)×L (− ,) + kv3∗ − v3 kL (ω ) ].
∼
∼
∼
∼
2
2
2
2
∼
∼
∼
∼
We will show that kσ 0 kLsym (ω ) and kσ 2 kLsym (ω ) are so small in the case of flexural
∼ ∼2
∼ ∼2
shells that we can totally give the factor to the second half of the above right hand
179
side. Using Theorem 6.3.5 and Cauchy’s inequality to the identity (6.3.17), we get
Z
Ω
Aijkl (σ kl − σ ∗kl )(σ ij − σ ∗ij ) +
Z
.
Ω
Z
Ω
C ijkl [χkl (v) − χkl (v ∗ )][χij (v) − χij (v ∗ )]
[Aijkl σ kl − χij (v)][σ ij − C ijkl χkl (v)] + kσ 0 k2 sym
∼
L2
∼
(ω )
+ kσ 2 k2 sym
∼
L
∼2
(ω )
.
(6.5.1)
6.5.1
Asymptotic behavior of the model solution
As we have seen in Chapter 3, if the shell is flexural, the model solution blows
up at the rate of O(−2 ). To get more accurate estimates, we need to scale the loading
functions by assuming
pio = 2 Poi ,
pie = 2 Pei ,
qai = 2 Qia ,
pi0 = 2 P0i ,
pi1 = 2 P1i ,
pi2 = 2 P2i ,
(6.5.2)
with Poi , Pei , Qia , P0i , P1i , and P2i independent of . Therefore, F 0 =−2 f 0 is a functional
independent of . Since we will consider the relative energy norm, this assumption is not
a requirement on the applied loads. It is just a technique to ease the analysis.
Under this scaling, the model solution ( θ , u , w ) converges to the solution
∼ ∼
( θ 0 , u 0 , w0 ) of the -independent limiting problem:
∼
∼
(ρ ( θ 0 , u 0 , w0 ), ρ (φ , y , z))U + hξ 0 , [γ (y , z), τ (φ , y , z)]i = hF 0 , (φ , y , z)i,
∼ ∼
∼
∼ ∼ ∼
∼ ∼
hη, [γ (u 0 , w0 ), τ ( θ 0 , u 0 , w0 )]i = 0,
∼ ∼
∼ ∼
∼
∼ ∼ ∼
∀ (φ , y , z) ∈ H, ∀ η ∈ W ∗ ,
∼ ∼
( θ 0 , u 0 , w0 ) ∈ H, ξ 0 ∈ W ∗ .
∼
∼
∼ ∼
(6.5.3)
180
This problem has a unique solution ( θ 0 , u 0 , w0 ) ∈ ker B, ξ 0 ∈ W ∗ . It is important to
∼
∼
note that ρ ( θ 0 , u 0 , w0 ) 6= 0. Otherwise, ( θ 0 , u 0 , w0 ) = 0, which is contradicted to the
∼ ∼
∼
∼
∼
flexural assumption f 0 |ker B 6= 0. From (3.3.5) of Chapter 3, we have
k( θ 0 , u 0 , w0 )kH + kξ 0 kW ∗ ' kF 0 kH ∗ .
∼
∼
The equation (6.5.3) is the limiting flexural shell model. This equation and its
solution provide indispensable supports to the ensuing analysis. For brevity, we denote
ρ0αβ = ραβ ( θ 0 , u 0 , w0 ).
∼
∼
Without any assumption on the regularity of the Lagrange multiplier ξ 0 ∈ W ∗
defined in the limiting problem (6.5.3), according to Theorem 3.3.3 and (3.4.5), we have
the strong convergence
kρ − ρ 0 kLsym (ω) + −1 kγ kLsym (ω) + −1 kτ kL (ω) → 0 ( → 0).
∼
∼ ∼2
∼ ∼2
∼ ∼2
(6.5.4)
If we assume more regularity on ξ 0 , say,
ξ 0 ∈ [V ∗ , W ∗ ]1−θ,q
(6.5.5)
for some θ ∈ (0, 1) and q ∈ [1, ∞] or θ ∈ [0, 1] and q ∈ (1, ∞), by Theorem 3.3.2 and
(3.4.4), we have
kρ − ρ 0 kLsym (ω) + −1 kγ kLsym (ω) + −1 kτ kL (ω) . K(, ξ 0 , [W ∗ , V ∗ ]) . θ .
∼
∼ ∼2
∼ ∼2
∼ ∼2
(6.5.6)
181
Recall that the K-functional on the Hilbert couple [W ∗ , V ∗ ], see [9], is defined as
K(, ξ 0 , [W ∗ , V ∗ ]) ' kξ 0 kW ∗ + V ∗ '
inf
ξ 0 =ξ10 +ξ20
(kξ10 kW ∗ + kξ20 kV ∗ ).
(6.5.7)
Based on the requirements imposed on the correction functions w1 and w2 , and
recalling the expressions (6.3.31) and (6.3.32) which need to be small, we define
w1 = 0
(6.5.8)
and define w2 as the solution of
λ
2 (∇w2 , ∇v)L (ω) + (w2 , v)L (ω) = −
(aαβ ρ0αβ , v)L (ω) ,
2
2
2
2(2µ + λ)
∼
(6.5.9)
1 (ω), ∀ v ∈ H 1 (ω).
w2 ∈ HD
D
The right hand side of the equation (6.5.9) is not a trivial extension of its analogue
in the Reissner–Mindlin plate theory developed in [2], according to which, ραβ , rather
than ρ0αβ , would have been used. We make this choice not only because of lack of
regularity of the dependent model solution, this choice of the correction functions
is also sufficient for us to prove the convergence and determine the convergence rate
in the next two subsections. The physical meaning of (6.5.8) is that, in the flexural
dominating deformation, the change of the shell thickness is negligible. In contrast,
the relative motion of the location of the middle surface is significant. For example, if
locally, the shell were bent down, the middle point would move toward the upper surface
and vise versa. The existence of such correction functions such that the convergence
182
can be proved is a sufficient justification of the model. Note that the correction does
not affect the middle surface deformation, which has already been well captured by the
model solution. In the forthcoming analysis of membrane–shear shells, we will choose
the opposite, w2 = 0.
Since the solution of the limiting problem (6.5.3) always guarantees that ρ 0 ∈
∼
sym
L
(ω), we have aαβ ρ0αβ ∈ L2 (ω). By (3.3.38) in Theorem 3.3.6, we have
∼2
kw2 kH 1 (ω) + k − w2 −
λ
aαβ ρ0αβ kL (ω) → 0 ( → 0).
2
2(2µ + λ)
(6.5.10)
If we assume
1 (ω), L (ω)]
aαβ ρ0αβ ∈ [HD
2
1−θ,p
(6.5.11)
for some θ ∈ (0, 1) and p ∈ [1, ∞], or θ ∈ [0, 1] and p ∈ (1, ∞), by (3.3.36) in Theorem 3.3.6, we have
kw2 kH 1 (ω) + k − w2 −
λ
1 (ω)]) . θ .
aαβ ρ0αβ kL (ω) . K(, aαβ ρ0αβ , [L2 (ω), HD
2
2(2µ + λ)
(6.5.12)
Remark 6.5.1. Both the assumptions (6.5.5) and (6.5.11) are requirements on regularity of the solution of the independent limiting problem (6.5.3), which are indirect
requirements on the shell data. The explicit dependence of the indices on these data
needs more analysis. The value of the index θ in (6.5.5) may be different from that in
(6.5.11). We choose the least one so that both the estimates (6.5.6) and (6.5.12) hold
simultaneously.
183
The asymptotic behavior of the model solution described by (6.5.4) and (6.5.6)
together with the equivalency (6.2.8) tell us that under the scaling of the loading functions
(6.5.2), the H 1 norm of the model solution is uniformly bounded:
k θ kH 1 (ω) . 1,
∼
∼
ku kH 1 (ω) . 1,
∼
∼
kw kH 1 (ω) . 1,
while the following estimate (6.5.18) shows that the strain energy engendered by the this
displacement is only of order O(3 ). This is the magnitude of strain energy that flexural
shells could sustain without collapsing.
6.5.2
Convergence theorems
As in Chapter 5, we denote the energy norms of a stress field σ and a strain field
χ on the shell Ω by kσkE and kχkE , which are equivalent to the sums of the L2 (ω )
norms of the tensor components.
Without making any assumption further than (6.2.9), it can be proved that the
model solution converges to the 3D solution in the relative energy norm. We have
Theorem 6.5.1. Let v ∗ and σ ∗ be the solution of the 3D shell problem, v the displacement field defined by the solution ( θ , u , w ) of the model (6.2.4) together with
∼ ∼
the correction functions w1 and w2 defined in (6.5.8) and (6.5.9) through the formulae
(6.3.13), and σ the stress field defined by (6.3.7). We have the convergence
kσ ∗ − σkE + kχ(v ∗ ) − χ(v)kE = 0.
kχ(v)kE →0
lim
(6.5.13)
184
If the solution ( θ 0 , u 0 , w0 ) and ξ 0 of the -independent limiting problem (6.5.3)
∼
∼
satisfies the condition
ξ 0 ∈ [V ∗ , W ∗ ]1−θ,q ,
1 (ω), L (ω)]
aαβ ρ0αβ ∈ [HD
2
1−θ,p
(6.5.14)
for some θ ∈ (0, 1) and p, q ∈ [1, ∞] or θ ∈ [0, 1] and p, q ∈ (1, ∞), we have the two
estimates (6.5.6) and (6.5.12) hold simultaneously, and we have
Theorem 6.5.2. If the regularity condition (6.5.14) is satisfied for some θ, we have the
convergence rate
kσ ∗ − σkE + kχ(v ∗ ) − χ(v)kE . θ .
kχ(v)kE (6.5.15)
We give the proof of Theorem 6.5.2. The proof of Theorem 6.5.1 is similar.
Proof. The proof is based on the inequality (6.5.1), the above two estimates (6.5.6)
αβ
and (6.5.12), the inequality (6.3.16) to bound σ2 , the expressions (6.3.28), (6.3.29) and
(6.3.30) for the constitutive residual %ij , and the scaling on the loads (6.5.2). In the
proof, the norm k · kL (ω ) will be simply denoted by k · k. Any function defined on ω
2
will be viewed as a function, constant in t, defined on ω .
First, we establish the lower bound for kχ(v)k2E . By the estimate (6.5.6), we
have
kρ − ρ 0 kLsym (ω) . θ ,
∼
∼ ∼2
kγ kLsym (ω) . 1+θ ,
∼
∼2
kτ kL (ω) . 1+θ ,
∼
2
(6.5.16)
∼
so
kρ kLsym (ω) ' kρ 0 kLsym (ω) ' 1.
∼
∼2
∼
∼2
(6.5.17)
185
From the equivalence (6.2.8), we see
k( θ , u , w )kH 1 (ω)×H 1 (ω)×H 1 (ω) ' kρ 0 kLsym (ω) ' 1.
∼ ∼
∼
∼
∼
∼2
The convergence (6.5.12) shows that kw2 kL (ω) ' kaαβ ρ0αβ kL (ω) . 1. Recalling the
2
2
expression (6.3.24)
1 2 γ γ 2
+ tρ − t(bλ γ + bλ γ ) + (tc
χαβ (v) = γαβ
α λβ
αβ − bαβ )t w2 − 2 t (bα θγ|β + bβ θγ|α ),
β λα
αβ
we can see that the dominant term in the right hand side of this equation is tραβ .
Therefore
2
X
kχαβ (v)k2 & 3 kρ kLsym (ω) .
∼
α,β=1
∼2
We obtain
kχ(v)k2E & 3 .
(6.5.18)
We then derive the upper bound on kσ ∗ − σk2E + kχ(v ∗ ) − χ(v)k2E . By the
inequality (6.5.1), we have
kσ ∗ −σk2E +kχ(v ∗ )−χ(v)k2E .
3
X
i,j=1
k%ij k2 +
2
X
2
X
αβ 2
αβ
kσ0 k +
kσ2 k2 . (6.5.19)
α,β=1
α,β=1
186
From the equations (6.3.1), we have
2
λ
αβ
+
= H 2 σ1 + aαβλγ γλγ
2 Po3 aαβ ,
3
2µ + λ
αβ
σ0
αβ
σ1
= aαβλγ ρλγ +
λ
2 (Pe3 + 2HPo3 )aαβ ,
2µ + λ
5
σ03α = [µaαβ τβ − 2 Poα ],
4
and so, we have the estimates
αβ
kσ1 k2 . ,
αβ
kσ0 k2 . 3+2θ ,
kσ03α k2 . 3+2θ .
(6.5.20)
By the estimate (6.3.16), we have
αβ
kσ2 k2 . 3+2θ .
(6.5.21)
From the equations (6.3.5) and (6.3.6), we have
σ033 =
2
αβ
(b σ + 2 Peα |α − 2H 2 Q3a ),
2 αβ 1
1
2
αβ
β αγ
αβ
β αγ
σ133 = [bαβ ((σ0 − 2 dγ σ1 ) + bαβ (σ2 − 2 dγ σ1 )
2
3
3
+ 2 Poα |α + 2 Pe3 + (1 + 2 K) 2 Q3a ],
and so the estimates
kσ033 k2 . 5 ,
kσ133 k2 . 5+2θ .
(6.5.22)
187
Applying all the above estimates to the expression (6.3.28) of %αβ , it is readily
seen that the square integral over ω of every term is bounded by O(5 ), except the term
λγ
(aαλ − tbαλ )(aβγ − tbβγ )σ2 −
λ
λγ
g a σ ,
2µ + 3λ αβ λγ 2
whose square integral on ω , according to (6.5.21), is bounded by O(3+2θ ). Therefore
we have
k%αβ k2 . 3+2θ .
(6.5.23)
From the convergence (6.5.12), we know kw2 kH 1 (ω) = θ , so kt2 ∂α w2 k2 . 3+2θ .
Together with (6.5.20), we get
k%3α k2 . 3+2θ .
(6.5.24)
Our final concern is about %33 . In the expression (6.3.30), the first term is
1 3
1
αβ
αβ
(p − νaαβ σ0 ) − w1 = (2 Po3 − νaαβ σ0 )
E o
E
whose square integral over ω is bounded, according to (6.5.20), by O(3+2θ ). The
second term is, see (6.3.32),
t[
1 3
λ
λ
αβ
(pe − νaαβ σ1 ) − 2w2 ] = t[−2w2 −
aαβ ρ0αβ ] − t
aαβ (ραβ − ρ0αβ )
E
2µ + λ
2µ + λ
+ t 2 [
1
2Hλ2
Pe3 −
P 3 ].
2µ + λ
µ(2µ + λ)(2µ + 3λ) o
188
By the convergence (6.5.12) and the estimate (6.5.16), we easily see that the square
integral of this term on ω is bounded by O(3+2θ ). The last term in (6.3.30) is also
bounded, by using (6.5.21) and (6.5.22), by O(3+2θ ). We get
k%33 k2 . 3+2θ .
(6.5.25)
Therefore, by (6.5.20), (6.5.21), and (6.5.19), we have the upper bound
kσ ∗ − σk2E + kχ(v ∗ ) − χ(v)k2E . 3+2θ .
(6.5.26)
The conclusion of the theorem follows from the lower bound (6.5.18) and the upper
bound (6.5.26)
By replacing θ and 2θ with o(1) in this proof, we will obtain a proof of Theorem 6.5.1.
Remark 6.5.2. The estimate kτ kL (ω) . 1+θ in (6.5.16) together with the conver∼
2
∼
gence theorem furnishes a justification of the Kirchhoff–Love hypothesis in the case of
flexural shells.
6.5.3
Plate bending
If the shell is flat, the model (6.2.4) degenerates to the Reissner–Mindlin plate
bending and stretching models analyzed in [2]. The limiting problem (6.5.3) combines
the mixed formulation of the Kirchhoff–Love biharmonic plate bending model and the
189
limiting plate stretching model. Under the loading assumption of this section, the solution of the limiting stretching equation is u 0 = 0. If the plate is totally clamped, the
∼
solution of the limiting problem is given by, see [8],
E
θ 0 = −∇w0 , ξ 0 = [0,
∇∆w0 ],
∼
∼
12(1 − ν 2 ) ∼
with w0 given as the solution of the biharmonic equation. If the plate boundary is
smooth, or is a convex polygon, and the loading function is smooth enough, such that
sym
the regularity w0 ∈ H 3 holds, then we have ξ 0 ∈ L2 (ω) × L2 (ω), which is equivalent
∼
∼
to V ∗ . Therefore, the index θ determined from (6.5.5) is 1.
It is readily seen that aαβ ρ0αβ = −∆w0 ∈ H 1 (ω). By the standard cut-off
argument, it can be shown that the index value θ determined from (6.5.11) is at least
1/2. Taking the minimum of these two values, the index value in (6.5.14) is at least 1/2,
which gives the convergence rate of the Reissner–Mindlin plate bending model. This
rate has already been shown to be optimal, see [16]. Therefore, Theorem 6.5.2 gives the
best possible estimate for flexural shells.
Plate stretching and shear dominated plate bending are also special shell problems
which are second kind membrane–shear shells, and will be remarked in the last section
of this chapter.
6.6
Totally clamped elliptic shells
For totally clamped elliptic shells, convergence of the model solution toward the
3D solution in the relative energy norm can be proved under the loading assumption
190
(6.2.9). Convergence rate will be determined and attributed to the regularity of the
solution of the -independent limiting membrane shell model. This regularity is defined
in terms of interpolation spaces. The rate O(1/6 ) will be established if the shell data
are smooth enough in the usual Sobolev sense.
A shell Ω is elliptic, if its middle surface S is uniformly elliptic in the sense that
the Gauss curvature K is strictly positive. I.e., there exists a K0 > 0, such that K ≥ K0 .
For any given t ∈ (− , ), We define S(t) = {Φ(x , t)|x ∈ ω)}, which is a surface parallel
∼
∼
to the middle surface S and at the height t. Let K(t) be the Gauss curvature of S(t),
from (4.1.6). It is easy to see that K(t) = K/(1 − 2tH + t2 K). So if S is elliptic, S(t) is
elliptic if t is small enough.
We assume the shell is totally clamped, So ∂D ω = ∂ω and ∂D ω = ∂ω × [− , ].
The space H then is H 10 (ω) × H 10 (ω) × H01 (ω). Under some smoothness assumption on
∼
∼
the shell middle surface S, the following Korn-type inequality was established in [23] and
[19]: There exists a constant C such that for any u ∈ H 10 (ω), w ∈ L2 (ω)
∼ ∼
+ kwk2L (ω) ≤ Ckγ (u , w)k2 sym ,
ku k2 1
∼ ∼
∼ H (ω)
L2 (ω)
2
∼
(6.6.1)
∼
where γ (u , w) is the membrane strain engendered by the displacement uα aα + wa3 on
∼ ∼
the middle surface, see (6.2.3). It was shown in [58] that this inequality is valid only on
totally clamped elliptic shells. Applying this inequality to the surface S(t), we get
ku k2 1
+ kwk2L (ω) ≤ C(t)kχαβ (u , w)k2 sym .
∼ H (ω)
∼
L2 (ω)
2
∼
∼
191
χαβ (u , w) is the tangential part of the 3D strain χij engendered by a displacement whose
∼
restriction on S(t) is uα g α + wg 3 . If is small enough, this inequality uniformly holds
for all t ∈ [− , ].
Taking integration at both sides of the above inequality with respect to t, we
see that there exists a constant 0 > 0 such that if ≤ 0 , for any displacement field
v = vi ∈ H 1D (ω ), we have
2
X
α=1
+ kv3 k2L (ω ) .
kvα k2 1
H (ω)×L2 (− ,)
2
3
X
i,j=1
kχij (v)k2L (ω ) .
2
(6.6.2)
Comparing this inequality to that given in Theorem 6.3.5, which is valid for all shells,
we see that the particularity of totally clamped elliptic shells is remarkable.
As what we did for the flexural shells, we first resolve the term r in the identity
(6.3.17). Using the expression (6.3.18), we have
|r| . [kσ 0 kLsym (ω ) + kσ 2 kLsym (ω ) ][kv ∗ − v kH 1 (ω)×L (− ,) + kv3∗ − v3 kL (ω ) ].
∼
∼
∼
∼
2
2
2
2
∼
∼
∼
∼
The inequality (6.6.2) allows us to give the factor to the first half of the above right
hand side. Using the inequality (6.6.2) and Cauchy’s inequality to the identity (6.3.17),
we get
Z
Ω
Aijkl (σ kl − σ ∗kl )(σ ij − σ ∗ij ) +
Z
.
Ω
Z
Ω
C ijkl [χkl (v) − χkl (v ∗ )][χij (v) − χij (v ∗ )]
[Aijkl σ kl − χij (v)][σ ij − C ijkl χkl (v)] + 2 kσ 0 k2 sym
∼
L2
∼
(ω )
+ 2 kσ 2 k2 sym
∼
L
∼2
(ω )
.
(6.6.3)
192
6.6.1
Reformulation of the resultant loading functional
From the inequality (6.6.1), it is immediately seen that for totally clamped elliptic
shells, we have ker B = 0. Therefore, no matter what is the resultant loading functional
in the model, the shell problem can never be flexural. Theorem 3.3.4, Theorem 3.3.5,
(3.4.6), and (3.4.7) in Chapter 3 are the right tools to analyze the asymptotic behavior of
the model solution. According to the classification of Section 3.5, if the condition (6.6.4)
below is satisfied, the totally clamped elliptic shell problem is of membrane–shear.
Since ker B = 0 and B is surjective from H to W , by the closed range theorem,
there exists a ζ∗0 ∈ W ∗ , such that the leading term of the resultant loading functional
can be equivalently written as
hf 0 , (φ , y , z)i = hζ∗0 , B(φ , y , z)i ∀ (φ , y , z) ∈ H.
∼ ∼
∼ ∼
∼ ∼
We recall that without further assumption, the solution of the model problem is untractable. The condition we imposed in Chapter 3 is ζ∗0 ∈ V ∗ . Under this condition, the
loading functional can be further written as
hf 0 , (φ , y , z)i = hζ∗0 , B(φ , y , z)i = (ζ 0 , B(φ , y , z))V ,
∼ ∼
∼ ∼
∼ ∼
(6.6.4)
here, ζ 0 ∈ V is the Riesz representation of ζ∗0 ∈ V ∗ . Therefore the condition (6.6.4) is
equivalent to the existence of (γ 0 , τ 0 ) ∈ V = V0 × L2 (ω), such that
∼ ∼
∼
Z
√
0 γ (y , z)√adx + 5 µ
aαβλγ γλγ
aαβ τβ0 τα (φ , y , z) adx . (6.6.5)
αβ ∼
∼ 6
∼ ∼
∼
ω
ω
Z
hf 0 , (φ , y , z)i =
∼ ∼
193
Recall that the expression of the leading term in the loading functional is
Z
Z
√
√
5
λ
α
po τα (φ , y , z) adx −
p3o aαβ γαβ (y , z) adx
hf 0 , (φ , y , z)i =
∼
∼
∼
∼ ∼
∼
∼
6 ω
2µ + λ ω
Z
α
3
3 √
α
α γ
+ [(pα
e + qa − 2bγ po )yα + (po |α + pe + qa )z] adx . (6.6.6)
∼
ω
Comparing this expression to (6.6.5), we just need to choose
τα0 =
1
β
a p ,
µ αβ o
(6.6.7)
obviously, τ 0 ∈ L2 (ω).
∼
∼
Thanks to the inequality (6.6.1), we know that γ defines an isomorphism between
∼
sym
the space H 10 (ω)×L2 (ω) and a closed subspace of L2 (ω), which should be V0 . Since the
∼
∼
last two terms in (6.6.6) together define a continuous linear functional on H 10 (ω)×L2 (ω),
∼
by the Riesz representation theorem, there exists a unique (u 0 , w0 ) ∈ H 10 (ω) × L2 (ω)
∼
∼
such that
γ 0 = γ (u 0 , w0 ) ∈ V0
∼
∼ ∼
(6.6.8)
and
Z
ω
√
aαβλγ γλγ (u 0 , w0 )γαβ (y , z) adx = −
∼
∼
Z
+
ω
∼
Z
√
λ
p3o aαβ γαβ (y , z) adx
∼
∼
2µ + λ ω
α
α γ
α
3
3 √
[(pα
e + qa − 2bγ po )yα + (po |α + pe + qa )z] adx
∼
∀ (y , z) ∈ H 10 (ω) × L2 (ω). (6.6.9)
∼
∼
194
Therefore, (6.6.7) and (6.6.8) together reformulated the resultant loading functional in
the desired way (6.6.5).
Note that the equation (6.6.9) can be viewed as an equation to determine the
functions (u 0 , w0 ) ∈ H 10 (ω) × L2 (ω). This is formally the same as the limiting elliptic
∼
∼
membrane shell model of [18], but note that the right hand side is different. Here, the
odd part of the surface force pio is incorporated.
6.6.2
Asymptotic behavior of the model solution
From Theorem 3.3.5 and (3.4.7), we get the asymptotic behavior of the model
solution ( θ , u , w ):
∼ ∼
kρ kLsym (ω) + kγ − γ 0 kLsym (ω) + kτ − τ 0 kL (ω) → 0
∼ ∼2
∼
∼ ∼2
∼
∼ ∼2
∼
( → 0).
(6.6.10)
∼
If we assume more regularity on (γ 0 , τ 0 ), say,
∼
∼
(γ 0 , τ 0 ) ∈ [W, V ]1−θ,q
∼
∼
(6.6.11)
for some θ ∈ (0, 1) and q ∈ [1, ∞], or θ ∈ [0, 1] and q ∈ (1, ∞), by Theorem 3.3.4 and
(3.4.6), we get the stronger estimate of the asymptotic behavior of the model solution:
kρ kLsym (ω) + kγ − γ 0 kLsym (ω) + kτ − τ 0 kL (ω) . K(, (γ 0 , τ 0 ), [V, W ]) . θ .
∼
∼
∼
∼
∼
∼ ∼
2
∼2
∼2
∼
(6.6.12)
We assume that γ 0 and τ 0 can not be zero simultaneously, otherwise f 0 = 0.
∼
∼
Under some further assumptions on the smoothness of loading functions, with a similar
195
but more tedious analysis, we can prove the convergence of the model solution to the 3D
solution even if f 0 = 0.
Based on the requirements imposed on the correction functions w1 and w2 , and
recalling the expressions (6.3.31) and (6.3.32), which need to be small, we define w1 as
the solution of the equation
λ
1
0 +
2 (∇w1 , ∇v)L (ω) + (w1 , v)L (ω) = (−
aαβ γαβ
p3 , v)
,
2
2
2µ + λ
2µ + λ o L2 (ω)
∼
(6.6.13)
w1 ∈ H01 (ω), ∀ v ∈ H01 (ω)
and define
w2 = 0.
(6.6.14)
The explanation of this choice of the correction functions is right in contrast to
what we made for flexural shells on page 181.
sym
0 ∈ L (ω).
From the definition (6.6.8) of γ 0 , we see that γ 0 ∈ L2 (ω), so aαβ γαβ
2
∼
∼
∼
By (3.3.38) in Theorem 3.3.6, we have the convergence
kw1 kH 1 (ω) + k − w1 −
λ
1
0 +
aαβ γαβ
p3 k
→ 0 ( → 0).
2µ + λ
2µ + λ o L2 (ω)
(6.6.15)
If we assume
0 − p3 ∈ [H 1 (ω), L (ω)]
λaαβ γαβ
2
o
1−θ,p
0
(6.6.16)
196
for some θ ∈ (0, 1) and p ∈ [1, ∞], or θ ∈ [0, 1] and p ∈ (1, ∞), by (3.3.36) in Theorem 3.3.6, we have,
kw1 kH 1 (ω) + k − w1 −
1
λ
0 +
aαβ γαβ
p3 k
2µ + λ
2µ + λ o L2 (ω)
0 − p3 , [L (ω), H 1 (ω)]) . θ . (6.6.17)
. K(, λaαβ γαβ
2
o
0
The values of the index θ in (6.6.11) and (6.6.16) might be different. We choose
the least one so that the convergences (6.6.12) and (6.6.17) hold simultaneously.
From the asymptotic estimates (6.6.10) we see
kρ kLsym (ω) . o(−1 ),
∼
∼2
kγ kLsym (ω) . 1,
∼
∼2
kτ kL (ω) . 1.
∼
2
∼
Under the regularity assumption (6.6.11), we have
kρ kLsym (ω) . θ−1 ,
∼
∼2
kγ kLsym (ω) . 1,
∼
∼2
kτ kL (ω) . 1.
∼
2
∼
By the equivalency (6.2.8) and the inequality (6.6.1), we get the a priori estimates
ku kH 1 (ω) . 1,
∼
kw kL (ω) . 1,
2
∼
k θ kH −1 (ω) . 1,
∼
∼
k θ kH 1 (ω) . o(−1 ) or O(θ−1 ), if (6.6.11),
∼
∼
kw kH 1 (ω) . k θ kL (ω) + ku kL (ω) .
∼
∼
2
2
∼
∼
(6.6.18)
197
6.6.3
Convergence theorems
The convergence of the 2D model solution to the 3D solution can be proved if
the loading functions satisfy the condition (6.2.9). Under further assumption on the
regularity of (γ 0 , τ 0 ), convergence rate can be established.
∼
∼
Theorem 6.6.1. Let v ∗ and σ ∗ be the 3D solution of the shell problem, v the displacement defined by the model solution ( θ , u , w ) together with the correction functions
∼ ∼
w1 , w2 defined in (6.6.13) and (6.6.14) through the formulae (6.3.13), and σ the stress
field defined by (6.3.7). Under the condition (6.2.9), we have the convergence
kσ ∗ − σkE + kχ(v ∗ ) − χ(v)kE = 0.
kχ(v)kE →0
lim
(6.6.19)
1
β
If γ 0 = γ (u 0 , w0 ) and τα0 = aαβ po satisfy the regularity condition
∼
∼ ∼
µ
(γ 0 , τ 0 ) ∈ [W, V ]1−θ,q
∼
∼
0 − p3 ∈ [H 1 (ω), L (ω)]
and λaαβ γαβ
2
o
1−θ,p
0
(6.6.20)
for some θ ∈ (0, 1) and p, q ∈ [1, ∞], or θ ∈ [0, 1] and p, q ∈ (1, ∞), then the convergences
(6.6.12) and (6.6.17) hold simultaneously, and we have
Theorem 6.6.2. If the regularity condition (6.6.20) is satisfied, we have the convergence
rate
kσ ∗ − σkE + kχ(v ∗ ) − χ(v)kE . θ .
kχ(v)kE (6.6.21)
With all the preparations of the last subsection, the proofs of these theorems are
almost the same as that of Theorem 5.5.1 for spherical shells, except that now we need
198
to use the inequality (6.6.3) rather than the two energies principle. For this reason, and
for lack of space, the proofs are omited.
The regularity condition (6.6.20) is not easy to interpret. We just give an unrealistic example to explain its meaning. We will determine the index θ in the next
subsection under some smoothness assumption on the shell data in the usual sense.
Let the shell be loaded in such a special way that in the reformulation of the
loading functional (6.6.5), γ 0 and τ 0 are given by
∼
0 = γ (u ◦ , w◦ )
γαβ
αβ ∼
∼
and
1
β
aαβ po = τα0 = τα ( θ ◦ , u ◦ , w◦ ),
∼ ∼
µ
with θ ◦ ∈ H 10 (ω), w◦ ∈ H01 (ω), and u ◦ ∈ H 10 (ω) ∩ H 2 (ω). We assume p3o ∈ H 1 (ω). It
∼
∼
∼
∼
∼
is easy to see that (γ 0 , τ 0 ) ∈ W , so the index θ determined from (6.6.11) is equal to 1.
∼
∼
0 − p3 ∈ H 1 (ω), by the standard cut-off argument, the index θ determined
Since λaαβ γαβ
o
from (6.6.16) is at least 1/2. The convergence rate then is determined by the smaller
one of these two values. I.e., at least 1/2 .
6.6.4
Estimates of the K-functional for smooth data
We have seen in the last subsection that the convergence rate of the model solution
to the 3D solution in the relative energy norm is determined by the the values of the
K-functionals in (6.6.12) and (6.6.17). In this subsection we estimate these values for the
elliptic shell under the assumption that the shell boundary, middle surface, and loading
functions are smooth enough.
199
Based on the definitions of the spaces W , V , and the K-functional (6.5.7), the
two K-functionals involved in the convergence can be equivalently expressed as
K(, (γ 0 , τ 0 ), [V, W ]) =
∼
=
∼
inf
(γ ,τ )∈W
[k(γ 0 − γ , τ 0 − τ )kV + k(γ , τ )kW ]
∼∼
∼
∼ ∼
∼
∼ ∼
[k(γ 0 − γ (u , w), τ 0 − τ ( θ , u , w))kLsym (ω)×L (ω) + k( θ , u , w)kH ]
∼ ∼
∼
∼ ∼ ∼
∼ ∼
2
2
( θ ,u ,w)∈H ∼
inf
∼
∼∼
∼
(6.6.22)
and
0 −p3 , [L (ω), H 1 (ω)]) =
K(, λaαβ γαβ
2
o
0
inf
0 −p3 −wk
[kλaαβ γαβ
o
L2 (ω) + kwkH 1 (ω) ].
w∈H01 (ω)
0
(6.6.23)
The strategy to determine the K-functional values is to make a good choice for
( θ , u , w) ∈ H 10 (ω) × H 10 (ω) × H01 (ω) in the former and a good choice for w ∈ H01 (ω) in
∼ ∼
∼
∼
the latter so that the infimums can be roughly reached. This can be done by doing a
little more delicate cut-off argument, which requires some regularity results.
We assume the following smoothness on the shell data: The shell boundary γ =
3
α
1
3
2
3
2
∂S ∈ C 4 . The loading functions pα
o ∈ H (ω), pe ∈ H (ω), po ∈ H (ω), pe ∈ H (ω),
qaα ∈ H 1 (ω), qa3 ∈ H 2 (ω).
Lemma 6.6.3. Under this assumption, the solution of the equation (6.6.9) has the regularity
u 0 ∈ H 3 (ω) ∩ H01 (ω)
∼
and
w0 ∈ H 2 (ω).
This lemma follows from a more general regularity theorem on the solution of the limiting
membrane shell model in [27]. Under the above smoothness assumption on the data, we
200
have the regularity
0 = γ (u 0 , w0 ) ∈ H 2 (ω),
γαβ
αβ ∼
τα0 =
1
β
a p ∈ H 3 (ω),
µ αβ o
0 − p3 ∈ H 2 (ω).
λaαβ γαβ
o
(6.6.24)
We also need the following cut-off lemma.
Lemma 6.6.4. Let ω ⊂ R2 be an open connected domain, and ∂ω ∈ C 2 . Let α > 0 and
> 0 be two positive numbers. Then for any f ∈ H 1 (ω), there exists a f 0 ∈ H01 (ω) such
that
kf − f 0 kL (ω) ≤ α kf kL (ω) , kf 0 kH 1 (ω) ≤ −α kf kH 1 (ω) .
2
2
If f ∈ H 2 (ω), we further have kf 0 kH 2 (ω) ≤ −3α kf kH 2 (ω) .
The proof of this lemma can be found in [36]. An equivalent result can be found in [43].
With these preparations, we can prove
Theorem 6.6.5. Under the above smoothness assumption on the shell data, we have the
following estimates on the K-functionals:
K(, (γ 0 , τ 0 ), [V, W ]) . 1/6
(6.6.25)
0 − p3 , [L (ω), H 1 (ω)]) . 1/2 .
K(, λaαβ γαβ
2
o
0
(6.6.26)
∼
∼
and
Therefore the value of the index θ is at least 1/6, which gives, by Theorem 6.6.2, the
convergence rate of the shell model.
201
Proof. According to (6.6.22), we need to estimate
[k(γ (u 0 , w0 ) − γ (u , w), τ 0 − τ ( θ , u , w))kLsym (ω)×L (ω) + k( θ , u , w)kH ].
∼ ∼
∼
∼ ∼ ∼
∼ ∼
2
2
( θ ,u ,w)∈H ∼ ∼
inf
∼
∼∼
∼
Since u 0 ∈ H 10 (ω), we can choose u = u 0 , so we have
∼
∼
∼
∼
γαβ (u 0 , w0 ) − γαβ (u , w) = bαβ (w − w0 ).
∼
∼
Taking a positive number a, since w0 ∈ H 2 (ω), by Lemma 6.6.4, there exists a
w ∈ H01 (ω) ∩ H 2 (ω) such that
kw0 − wkL (ω) ≤ a kw0 kL (ω) ,
2
2
kwkH 1 (ω) ≤ −a kw0 kH 1 (ω) ,
kwkH 2 (ω) ≤ −3a kw0 kH 2 (ω) .
From the definition (6.2.3), we have τα ( θ , u , w) = θα + ∂α w + bλ
α uλ . Let b be
∼ ∼
a positive numberi. By Lemma 6.6.4, for the above chosen u and w, there exists a
∼
θ ∈ H 10 (ω) such that
∼
∼
kτ 0 − τ ( θ , u , w)kL (ω) '
∼
∼ ∼ ∼
2
∼
2
X
α=1
≤ b
0
kθα + ∂α w + bλ
α uλ − τα kL2 (ω)
2
X
α=1
0
k∂α w + bλ
α uλ − τα kL2 (ω)
≤ b−a kw0 kH 1 (ω) + b ku 0 kL (ω) + b kτ 0 kL (ω)
∼
∼
2
2
∼
∼
202
and
k θ kH 1 (ω) ≤ −b
∼ ∼
2
X
0
k∂α w + bλ
α uλ − τα kH 1 (ω)
α=1
≤ −b−3a kw0 kH 2 (ω) + −b ku 0 kH 1 (ω) + −b kτ 0 kH 1 (ω) .
∼
∼
∼
∼
With these ( θ , u , w) substituted in the arguments of the infimum, we see
∼ ∼
K(, (γ 0 , τ 0 ), [V, W ]) . a kw0 kL (ω) + b−a kw0 kH 1 (ω) + b ku 0 kL (ω)
∼ ∼
∼
2
2
∼
+ b kτ 0 kL (ω) + 1−b−3a kw0 kH 2 (ω) + (1−b + )ku 0 kH 1 (ω)
∼
∼
2
∼
∼
+ 1−b kτ 0 kH 1 (ω) + 1−a kw0 kH 1 (ω) .
∼
∼
Note that τ 0 , u 0 and w0 are all -independent functions. The best values for a and b
∼
∼
should make a = b − a = 1 − b − 3a, and are given by a = 1/6, b = 1/3. We obtain
K(, (γ 0 , τ 0 ), [V, W ]) . 1/6 .
∼
∼
(6.6.27)
The proof of (6.6.26) is simpler and so ignored.
Based on this estimate, we can compare the strain energy that can be sustained by
a totally clamped spherical shell with that which can be sustained by a totally clamped
flexural plate. The former is a special totally clamped elliptic shell, and the latter is a
special flexural shell. For the plate, by (6.5.18), the strain energy is O(3 ), and the model
solution tends to a finite limit in the space H. For spherical shell, by (5.5.24), the strain
203
energy is O(), but by the estimate (6.6.18), the H norm of the solution is only bounded
by O(−5/6 ). To keep the solution bounded, we have to reduce the loads by multiplying a
factor of O(5/6 ). The strain energy will be scaled to O([5/6 ]2 ) = O(8/3 ). Therefore,
without blowing up in displacement, the strain energy that can be sustained by a totally
clamped spherical shell is O(−1/3 ) times that can be sustained by a plate.
1
β
If τα0 = aαβ po = 0, we can make another choice of ( θ , u , w) ∈ H in the proof
∼ ∼
µ
of Theorem 6.6.5 and prove
K(, (γ 0 , 0), [V, W ]) . 1/5 .
(6.6.28)
∼
This can be done by letting u = u 0 , choosing w ∈ H02 (ω) such that
∼
∼
kw − w0 kL (ω) . 1/5 ,
2
kwkH 1 (ω) . −1/5 ,
kwkH 2 (ω) . −4/5 ,
and taking θα = −∂α w − bλ
α uλ . The existence of such w can be proved by using a lemma
of [45]. Therefore, if the odd part of the tangential surface forces vanishes, the model
convergence rate is O(1/5 ) in the relative energy norm.
The estimate (6.6.12) not only plays a crucial role in establishing the convergence
rate of the model, but also gives an estimate on the difference between our model solution
and the solution (u 0 , w0 ) of the limiting model (6.6.9). The K-functional value will be
∼
used to prove the convergence rate of the limiting model solution in Section 7.4.
As we have mentioned at the end of the last subsection, the convergence rate can
be as high as 1/2 , but we can only prove this when the loading functions are special.
204
If we only assume the smoothness on the shell data in the usual Sobolev sense, under
the most general loading assumption, the convergence rate O(1/6 ) is the best we can
prove. It seems possible to get better results by other methods, see [33], [52], [53], [26],
and [39].
Remark 6.6.1. The convergence kτ − τ 0 kL (ω) → 0 ( → 0) in (6.6.10), or the
∼
∼
2
∼
estimate kτ − τ 0 kL (ω) . θ in (6.6.12), together with the expression (6.6.7) and
∼
∼ ∼2
Theorems 6.6.1 and 6.6.2 violate the Kirchhoff–Love hypothesis if the odd part of the
tangential surface force pα
o is not zero. This is in sharp contrast to the case of flexural
shells for which the Kirchhoff–Love assumption can always be proved, see Remark 6.5.2.
On the other hand, these convergences furnish a proof for this hypothesis if, say, the odd
part of the tangential surface force vanished, cf., [40].
6.7
Membrane–shear shells
The totally clamped elliptic shells we discussed in the previous section are special
examples of general membrane–shear shells defined in Section 3.5 and Section 6.4. There
are two major difficulties in general situation. One lies in the reformulated resultant
loading functional:
hf 0 , (φ , y , z)i = hζ∗0 , B(φ , y , z)i ∀ (φ , y , z) ∈ H,
∼ ∼
∼ ∼
∼ ∼
(6.7.1)
with ζ∗0 ∈ W ∗ . To apply the abstract theory of Chapter 3 to analyze the asymptotic
behavior of the model solution, we need to assume ζ∗0 ∈ V ∗ . This condition, which
205
was unconditionally satisfied by totally clamped elliptic shells, now imposes a stringent
restriction on the resultant loading functional.
Another difficulty, which is even more formidable, lies in resolving the extra term
r in the integration identity (6.3.17). This identity, as in the last two sections, plays
the keystone role in the model justification. We can neither resolve this extra term in
the way of handling totally clamped elliptic shells, see (6.6.3), since the -independent
Korn-type inequality (6.6.2) is no longer valid, nor can we resort to the measure for
flexural shells, see (6.5.1), because the quantity kσ 0 k2 sym is not small any more.
∼ L2 (ω )
Both of these difficulties will be eluded by imposing further conditions. The
formulations and proofs of convergence theorems will otherwise be the same as those in
the last section. The shell problems that are ruled out by these conditions abound, for
which the convergence of the model solutions to the 3D solutions might not hold in the
relative energy norm.
6.7.1
Asymptotic behavior of the model solution
In this subsection, we interpret the abstractly imposed condition ζ∗0 ∈ V ∗ for
general membrane–shear shell problems, and analyze the asymptotic behavior of the
model solution by using the abstract theory of Chapter 3.
206
The condition ζ∗0 ∈ V ∗ is equivalent to the existence of (γ 0 , τ 0 ) ∈ V , such that
∼ ∼
ζ 0 = (γ 0 , τ 0 ), and
∼
∼
hf 0 , (φ , y , z)i = (ζ 0 , B(φ , y , z))V
∼ ∼
Z
=
ω
∼ ∼
0 γ (y , z)√adx + 5 µ
aαβλγ γλγ
αβ ∼
∼ 6
Z
ω
√
aαβ τβ0 τα (φ , y , z) adx ∀ (φ , y , z) ∈ H.
∼ ∼
∼
∼ ∼
(6.7.2)
Recalling the expression (6.2.5)
Z
Z
√
√
5
λ
α
hf 0 , (φ , y , z)i =
po τα (φ , y , z) adx −
p3o aαβ γαβ (y , z) adx
∼ ∼
∼
∼
∼
∼
∼
6 ω
2µ + λ ω
Z
Z
√
γ
α
α
α
α
3
3
+ [(pe + qa − 2bγ po )yα + (po |α + pe + qa )z] adx +
pα
0 yα (6.7.3)
∼
ω
γT
and the factorization (6.4.1) of the space V = V0 × L2 (ω), we see that the requirement
(6.7.2) is equivalent to the following two requirements. First,
pα
o ∈ L2 (ω)
and
p3o ∈ L2 (ω).
(6.7.4)
Second, there exists a κ ∈ V0 , such that
∼
Z
ω
aαβλγ κ
√
λγ γαβ (y , z) adx =
∼
∼
Z
ω
α
α γ
α
3
3 √
[(pα
e + qa − 2bγ po )yα + (po |α + pe + qa )z] adx
∼
Z
+
γT
1
1
pα
0 yα ∀ (y , z) ∈ H D (ω) × HD (ω). (6.7.5)
∼
∼
207
Note that the second term in the right hand side of (6.7.3) can be equally written
as
Z
Z
√
√
λ
λ
3
αβ
po a γαβ (y , z) adx = −
aαβλγ aλγ p3o γαβ (y , z) adx .
−
∼
∼
∼
∼
2µ + λ ω
2µ(2µ + 3λ) ω
(6.7.6)
Therefore, if p3o ∈ L2 (ω), we can determine γ 0 ∈ V0 as
∼
λ
0 =κ
3
γαβ
αβ − 2µ(2µ + 3λ) PV0 (aαβ po ),
sym
where PV0 is the orthogonal projection from L2
∼
τα0 =
(6.7.7)
(ω) to V0 . By defining
1
β
a p ,
µ αβ o
(6.7.8)
we obtain ζ 0 = (γ 0 , τ 0 ) ∈ V such that the loading functional be reformulated as (6.7.2).
∼
∼
Under the condition (6.7.2), the asymptotic behavior of the model solution then
follows from Theorem 3.3.5 and (3.4.7). We have
kρ kLsym (ω) + kγ − γ 0 kLsym (ω) + kτ − τ 0 kL (ω) → 0
∼ ∼2
∼
∼ ∼2
∼
∼ ∼2
∼
( → 0).
(6.7.9)
∼
If we assume more regularity on (γ 0 , τ 0 ), say,
∼
∼
(γ 0 , τ 0 ) ∈ [W, V ]1−θ,q
∼
∼
(6.7.10)
208
for some θ ∈ (0, 1) and q ∈ [1, ∞], or θ ∈ [0, 1] and q ∈ (1, ∞), by Theorem 3.3.5 and
(3.4.6), we have the stronger estimate of the asymptotic behavior of the model solution:
kρ kLsym (ω) + kγ − γ 0 kLsym (ω) + kτ − τ 0 kL (ω) . K(, (γ 0 , τ 0 ), [V, W ]) . θ .
∼ ∼2
∼
∼ ∼2
∼
∼ ∼2
∼ ∼
∼
∼
(6.7.11)
From the asymptotic estimate (6.7.9), we see
kρ kLsym (ω) . o(−1 ),
∼
∼2
kγ kLsym (ω) . 1,
∼
kτ kL (ω) . 1.
∼
2
∼
∼2
Under the regularity assumption (6.7.10), by (6.7.11), we have
kρ kLsym (ω) . θ−1 ,
∼
∼2
kγ kLsym (ω) . 1,
∼
∼2
kτ kL (ω) . 1.
∼
2
∼
By the equivalency (6.2.8), we get the a priori estimates
k θ kH 1 (ω) + ku kH 1 (ω) + kw kH 1 (ω) . o(−1 )
∼
∼
∼
∼
(or k θ kH 1 (ω) + ku kH 1 (ω) + kw kH 1 (ω) . θ−1 ,
∼
∼
∼
if the regularity (6.7.10) holds),
∼
kw kH 1 (ω) . k θ kL (ω) + ku kL (ω) .
∼
∼
2
2
∼
∼
(6.7.12)
These estimates are much weaker than those for totally clamped elliptic shells, see
(6.6.18), because of lack of the Korn-type inequality (6.6.1), which is a characterization of totally clamped elliptic shells.
209
The two conditions (6.7.4) and (6.7.5) together are equivalent to the condition
(6.7.2). The first condition (6.7.4) is trivially satisfied, while the second one (6.7.5), i.e.,
the existence of κ ∈ V0 such that (6.7.5) holds, can be connected to the “generalized
∼
membrane shell” theory, see [20] and [18], in the following way.
The membrane strain operator γ (y , z) defines a linear continuous operator
∼ ∼
1 (ω) −→ V ,
γ : H 1D (ω) × HD
0
∼
∼
whose range is dense in V0 . We first consider the case of ker γ = 0. In this case,
∼
1 (ω), which is weaker than the
kγ (y , z)kV0 defines a norm on the space H 1D (ω) × HD
∼ ∼
∼
1 (ω) with
original norm. In the notation of [18], we denote the completion of H 1D (ω)×HD
∼
]
]
respect to this new norm by VM (ω). Obviously, γ can be uniquely extended to VM (ω),
∼
and the extended linear continuous operator, still denoted by γ , defines an isomorphism
∼
]
]
between VM (ω) and V0 . By the closed range theorem, for any f ∈ [VM (ω)]∗ , there exists
a unique κ ∈ V0 , such that
∼
Z
ω
√
]
aαβλγ κλγ γαβ (y , z) adx = hf, (y , z)i ∀ (y , z) ∈ VM (ω).
∼
∼
∼
∼
(6.7.13)
Therefore, the problem of existence of κ ∈ V0 in (6.7.5) is equivalent to the problem
∼
that whether or not the linear functional
Z
ω
α
α γ
α
3
3 √
[(pα
+
e + qa − 2bγ po )yα + (po |α + pe + qa )z] adx
∼
Z
γT
pα
0 yα ,
210
1 (ω) by the right hand side of (6.7.5), can be
which is defined on the space H 1D (ω) × HD
∼
]
extended to a linear continuous functional on the space VM (ω).
]
The characterization of the space VM (ω) depends on the geometry of the shell
middle surface, shape of the lateral boundary, and type of lateral boundary condition.
If the shell is a totally clamped elliptic shell, by the inequality (6.6.1), it is easily
determined that
]
VM (ω) = H 10 (ω) × L2 (ω),
∼
and, as we have shown, the mild condition (6.2.9) is enough to guarantee the existence
of κ ∈ V0 such that (6.7.5) holds.
∼
If the shell is a stiff hyperbolic shell, it was shown in [44], see also [18], that
]
VM (ω) = a closed subspace of
L (ω) × H
∼2
−1 (ω),
therefore, the existence of κ ∈ V0 is guaranteed if
∼
γ
α
α
pα
e + qa − 2bγ po ∈ L2 (ω),
3
3
1
pα
o |α + pe + qa ∈ H0 (ω),
and pα
0 = 0.
(6.7.14)
If the shell is a stiff parabolic shell, it was shown in [44], see also [18], that
]
VM (ω) = a closed subspace of H −1 (ω) × H −2 (ω),
∼
so, the existence of κ ∈ V0 is guaranteed if
∼
γ
α
α
1
pα
e + qa − 2bγ po ∈ H0 (ω),
3
3
2
pα
o |α + pe + qa ∈ H0 (ω),
and pα
0 = 0.
(6.7.15)
211
]
If the shell is a partially clamped elliptic shell, the space VM (ω) can be huge and
its norm so weak, that the equation (6.7.5) may have no solution even if the loading
functions are in D(ω), the space of test functions of distribution. In this case, even if the
loading functions make the problem solvable, the problem can not afford an infinitesimal
smooth perturbation on the loads, see [38].
]
Since γ defines an isomorphism between VM (ω) and V0 , so the existence of κ ∈ V0
∼
∼
]
]
means the existence of (u , w) ∈ VM (ω)(the element in VM (ω) must be viewed as an
∼
entity, the notation in components might have no usual sense), such that κ = γ (u 0 , w0 ).
∼
∼ ∼
Therefore, the problem (6.7.5) of determining κ ∈ V0 is equivalent to finding (u 0 , w0 ) ∈
∼
∼
]
VM (ω), such that
Z
ω
√
aαβλγ γλγ (u 0 , w0 )γαβ (y , z) adx
∼
∼
Z
=
ω
∼
α
α γ
α
3
3 √
[(pα
e + qa − 2bγ po )yα + (po |α + pe + qa )z] adx
∼
Z
+
γT
]
pα
0 yα ∀ (y , z) ∈ VM (ω). (6.7.16)
∼
This variational equation is the same as the “generalized membrane shell” model, except
that in the formulation of the right hand side we incorporated the odd part of the surface
forces. Note that we are not looking for the solution of this generalized membrane shell
]
problem in the space VM (ω). Our interest is in the existence of κ ∈ V0 .
∼
The case of ker γ 6= 0 can be divided in two different kinds corresponding to
∼
ker B = 0 and ker B 6= 0. The first kind is the “second kind generalized membrane
shells” of [20] and [18]. The second kind is our second kind membrane–shear shells.
212
Since there is not an imaginable realistic example of the “second kind generalized membrane shell”, we will not discuss it in details here, but just remark that the
requirement (6.7.5), which leads to the convergence of the model solution to the 3D
solution in the relative energy norm, is not equivalent to the condition imposed in [18].
Our requirement is more restrictive, and might have excluded some situations analyzed
there. In other word, some of the “second kind generalized membrane shells” made the
equivalent representation ζ 0 of the resultant loading functional f 0 belong to (V ∗ , W ∗ ].
The Naghdi-type model is no longer membrane–shear dominated. Their analysis shows
that in some weak sense, and in a quotient space, it is still possible to replace the nonmembrane dominated problem by a membrane problem.
Examples for the second kind membrane–shear shells include plate stretching and
shear dominated plate bending, which have been thoroughly analyzed, and the condition
(6.7.5) does not impose a stringent restriction on the loading functions, see [2] and [5].
Another example is the membrane–shear cylindrical shell analyzed in Chapter 2, for
which, we have V = W , so the condition (6.7.5) is trivially satisfied.
Based on the requirements imposed on the correction functions w1 and w2 , and
the expressions (6.3.31) and (6.3.32), which need to be small, we define w1 as the solution
of the equation
λ
1
0 +
2 (∇w1 , ∇v)L (ω) + (w1 , v)L (ω) = (−
aαβ γαβ
p3 , v)
,
2
2µ + λ
2µ + λ o L2 (ω)
∼2
1 (ω), ∀ v ∈ H 1 (ω),
w1 ∈ HD
D
(6.7.17)
213
and define
w2 = 0.
(6.7.18)
The explanation of this choice of the correction functions is similar to that made for
totally clamped elliptic shell.
sym
From the definition (6.7.7) of γ 0 , we know γ 0 ∈ L2
∼
∼
∼
0 ∈
(ω), so we have aαβ γαβ
L2 (ω). By (3.3.38) in Theorem 3.3.6, we have
kw1 kH 1 (ω) + k − w1 −
λ
1
0 +
aαβ γαβ
p3 k
→ 0 ( → 0).
2µ + λ
2µ + λ o L2 (ω)
(6.7.19)
If we assume
0 − p3 ∈ [H 1 (ω), L (ω)]
λaαβ γαβ
2
o
1−θ,p
D
(6.7.20)
for some θ ∈ (0, 1) and p ∈ [1, ∞], or θ ∈ [0, 1] and p ∈ (1, ∞), by (3.3.36) in Theorem 3.3.6, we have
kw1 kH 1 (ω) + k − w1 −
λ
1
0 +
aαβ γαβ
p3 k
2µ + λ
2µ + λ o L2 (ω)
0 − p3 , [L (ω), H 1 (ω)]) . θ . (6.7.21)
. K(, λaαβ γαβ
2
o
D
6.7.2
Admissible applied forces
To prove the convergence theorem, in addition to the asymptotic behaviors of the
model solution (6.7.9) and (6.7.19), which hinge on the validity of the condition (6.7.5),
we also need to bound the term r in the right hand side of the integration identity
(6.3.17). Except for some special shells, like plates and spherical shells, the desirable
214
bound can only be obtained under some restrictions on the applied forces on the 3D
shell. As a sufficient condition, we adopt the condition of “admissible applied forces”
proposed in [18].
We recall that, the 3D shell Ω is subjected to body force and surface tractions
on the upper and lower surfaces Γ± , it is clamped along a part of its lateral surface ΓD ,
and loaded by a surface force on the remaining part of the lateral face ΓT . The body
force density is q = q i g i , the upper and lower surface force densities are p± = pi± g i , and
the lateral surface force density is pT = piT g i . To describe the concept of “admissible
applied forces”, we consider the work done by these applied forces over an admissible
displacement v = vi g i , which is given by
Z
L(v) =
Ω
q i vi +
Z
Γ±
pi vi +
Z
ΓT
piT vi .
(6.7.22)
Let v ∗ ∈ H 1D (ω ) be the displacement solution of the 3D shell problem. By
ij
adapting the notation of [18], we denote the actual stress distribution by F = σ ∗ij ∈
sym
L2
(ω ). Therefore,
Z
L(v) =
ω
√
ij
F χij (v) gdx dt.
∼
(6.7.23)
ij
We scale the 3D shell displacement v ∗i and the stress F from the coordinate
domain ω to the fat domain ω 1 , and denote the scaled displacement by v ∗ () and the
scaled stress by F ij (), by defining
t
v ∗ ()(x , ) = v ∗ (x , t),
∼ ∼
t
ij
F ij ()(x , ) = F (x , t), ∀ x ∈ ω, t ∈ (− , ).
∼ ∼
∼
(6.7.24)
215
In [18], the tensor valued function F ij () was directly introduced to reformulate
the linear form L(v) (6.7.22) in the form (6.7.23), but on the scaled domain ω 1 . The
connection between the tensor valued function F ij () and the actual stress distribution
over the loaded shell is our observation.
The applied forces are called admissible, if
sym
1. F ij () is uniformly bounded in L2
(ω 1 ) with respect to .
sym
2. There exists a tensor field F ij ∈ L2
(ω 1 ), independent of , such that
sym
lim F ij () = F ij in L2
→0
(ω 1 ),
(6.7.25)
see page 265 in [18].
Since F ij () is the actual stress distribution scaled to ω 1 , this condition also
implies the convergence of the scaled strain tensor χ (v ∗ ()) defined in (6.3.34). It
seems that this condition has assumed the convergence of the solution of the 3D shell
problem when → 0. But the question is how to identify the limit F ij . This limit can
only be correctly determined by resorting to a lower dimensional shell model. Therefore,
the shell theories established under the assumption of “admissible applied forces” is not
totally trivial.
From the first condition, we see that the scaled strain χij (v ∗ ()) of the shell
sym
deformation (see (6.3.33) and (6.3.34) for definition), is uniformly bounded in L2
(Ω).
By the Korn-type inequality on thin shells (6.3.35), we get the following bound on the
216
scaled displacement
kv ∗ ()k 1 1 . 1
H D (ω )
(6.7.26)
Under the second condition, we can extract a weak convergent subsequence from { v ∗ ()}
in H 1D (ω 1 ), then find the weak limit and pass to strong convergence, and finally prove
the following convergence,
lim kv ∗ ()k 1 1 = 0,
H D (ω )
→0
(6.7.27)
see [18] for details. Note that this behavior of the 3D shell solution is compatible with
the behavior (6.7.12) of the 2D model solution, yet another evidence for the necessity of
the assumption on the admissibility of the applied forces.
By rescaling the convergence (6.7.27) back to the domain ω , we will get
1/2 (
2
X
α=1
∗k
∗
kvα
H 1 (ω)×L2 (− ,) + kv3 kL2 (ω ) ) . o(1)
(6.7.28)
This inequality is what we need to prove our theorem.
6.7.3
Convergence theorem
For the general membrane–shear shells, under the condition (6.7.5) assumed on
the 2D model problem (6.2.4) and the condition (6.7.25) imposed on the 3D shell problem,
we have the convergence theorem:
Theorem 6.7.1. Let v ∗ and σ ∗ be the displacement and stress of the shell determined
from the 3D elasticity equations, v the displacement defined through the formulae (6.3.13)
217
in terms of the 2D model solution ( θ , u , w ) and the correction functions w1 , w2 de∼ ∼
fined in (6.7.17) and (6.7.18), and σ the stress field defined by (6.3.7). We have the
convergence
kσ ∗ − σkE + kχ(v ∗ ) − χ(v)kE = 0.
kχ(v)kE →0
lim
(6.7.29)
Proof. Except for the different way to bound the term r in the identity (6.3.17), the proof
is otherwise the same as that of the Theorems 6.6.1 and 5.5.1. The proof is based on
the identity (6.3.17), the inequality (6.7.28), the two convergences (6.7.9) and (6.7.19),
αβ
the inequality (6.3.16) to bound σ2 , and the expressions (6.3.28), (6.3.29) and (6.3.30)
for the constitutive residual %ij . Again, for brevity, the norm k · kL (ω ) will be simply
2
denoted by k · k. Any function defined on ω will be viewed as a function, constant in t,
defined on ω .
First, we establish the lower bound for the strain energy engendered by the displacement v. By the convergence (6.7.9), we have
kρ kLsym (ω) . o(1),
∼
∼2
kγ − γ 0 kLsym (ω) . o(1),
∼
∼
∼2
kτ − τ 0 kL (ω) . o(1). (6.7.30)
∼
∼
2
∼
Since γ 0 and τ 0 can not be zero at the same time (otherwise f 0 = 0), we have
∼
∼
kγ kLsym (ω) + kτ kL (ω) ' kγ 0 kLsym (ω) + kτ 0 kL (ω) ' 1.
∼ ∼2
∼ ∼2
∼ ∼2
∼ ∼2
∼
(6.7.31)
∼
By the equivalence (6.2.8), we have
k( θ , u , w )kH 1 (ω)×H 1 (ω)×H 1 (ω) . o(1).
∼ ∼
∼
∼
(6.7.32)
218
The convergence (6.7.19) shows
0 − p3 k
kw1 kH 1 (ω) . o(1) and kw1 kL (ω) ' kλaαβ γαβ
o L2 (ω) .
2
(6.7.33)
Recalling the expression (6.3.24), we have
1 2 γ γ + tρ − t(bλ γ + bλ γ ) + t(tc
χαβ (v) = γαβ
α λβ
αβ − bαβ )w1 − 2 t (bα θγ|β + bβ θγ|α )
αβ
β λα
and
1
1
χα3 (v) = τα + t∂α w1 ,
2
2
and
in which, by the estimates (6.7.30), (6.7.31), (6.7.32), and (6.7.33), the terms γαβ
τα dominate respectively. Summerizing these estimates, we get
2
X
kχαβ (v)k2 +
2
X
α=1
α,β=1
kχ3α (v)k2 & [kγ k2 sym
∼
L2
∼
(ω)
+ kτ k2L (ω) ] & .
∼
2
∼
Therefore,
kχ(v)k2E & .
(6.7.34)
We then derive the upper bound on kσ ∗ − σk2E + kχ(v ∗ ) − χ(v)k2E . From the
identity (6.3.17), we have
kσ ∗ − σk2E + kχ(v ∗ ) − χ(v)k2E .
3
X
i,j=1
k%ij k2 + |r|
(6.7.35)
219
From the expression (6.3.21) of r, we see
|r| . 2
X
αβ
αβ
(kσ0 k + kσ2 k)[
2
X
α=1
α,β=1
+ 5/2
2
X
∗k
∗
kvα
H 1 (ω)×L2 (− ,) + kv3 k]
αβ
αβ
(kσ0 k + kσ2 k)[k θ kH 1 (ω) + kw1 kL (ω) ]. (6.7.36)
∼
2
∼
α,β=1
From the equations (6.3.1) and (6.7.8), we have
αβ
σ0
λ
2
αβ
+
p3 aαβ ,
= H 2 σ1 + aαβλγ γλγ
3
2µ + λ o
αβ
σ1
= aαβλγ ρλγ +
λ
(p3 + 2Hp3o )aαβ ,
2µ + λ e
5
5
σ03α = [µaαβ τβ − pα
] = [µaαβ (τβ − τβ0 )],
o
4
4
and so, the estimates
αβ
2 kσ1 k2 . o(),
αβ
kσ0 k2 . ,
kσ03α k2 . o().
(6.7.37)
By the estimate (6.3.16), we have
αβ
kσ2 k2 . o().
(6.7.38)
Combining (6.7.37) and (6.7.38), we see
2
X
αβ
αβ
(kσ0 k + kσ2 k) . O(1/2 ).
α,β=1
(6.7.39)
220
Together with the inequality (6.7.28), we get the upper bound on the first term in the
right hand side of (6.7.36):
2
X
αβ
αβ
(kσ0 k + kσ2 k)[
2
X
α=1
α,β=1
∗k
∗
kvα
H 1 (ω)×L2 (− ,) + kv3 k] . o().
(6.7.40)
Using (6.7.32), (6.7.33) and (6.7.39), we get the bound on the second term:
5/2
2
X
αβ
αβ
(kσ0 k + kσ2 k)[k θ kH 1 (ω) + kw1 kL (ω) ] . o(2 ).
∼
2
(6.7.41)
∼
α,β=1
Therefore, we obtain
|r| . o()
(6.7.42)
The proof of
3
X
k%ij k2 . o()
i,j=1
is a verbatim repetition of the relevant part in the proof of Theorem 5.5.1. By (6.7.35),
we get
kσ ∗ − σk2E + kχ(v ∗ ) − χ(v)k2E . o().
The conclusion of the theorem follows from this inequality and the lower bound (6.7.34).
To get a convergence rate, we need to use the asymptotic behaviors (6.7.11) and
(6.7.21) of the model solution, whose validity depends on the assumptions (6.7.10) and
(6.7.20), and a more strict requirement on the applied forces on the 3D shell. Otherwise,
221
the statement of the theorem on the convergence rate is the same as Theorem 6.6.2, and
the proof would be a modification of that of Theorem 6.7.1.
The conclusion of this theorem seems stronger than other theories for the general
membrane–shear shells.
222
Chapter 7
Discussions and justifications
of other linear shell models
In this final chapter, we briefly discuss the justifications of some other linear
shell models based on the convergence theorems we proved for the model (6.2.4). The
discussion is in the context of Chapter 6. All these models can be viewed as variants of
the general shell model (6.2.4). We recall that the solution of the general shell model
(6.2.4) was denoted by ( θ , u , w ). In terms of this model solution and the transverse
∼ ∼
deflection correction functions w1 and w2 , we defined an admissible displacement field
v by the formulae (6.3.13). The convergence and convergence rate in the relative energy
norm of v toward the 3D displacement solution v ∗ were proved.
For each variant of the general shell model, we will re-define displacement functions ( θ̄ , ū , w̄ ) ∈ H from its solution. The the correction functions w1 and w2 will be
∼ ∼
defined either by (6.5.8) and (6.5.9) or by (6.7.17) and (6.7.18), depending on whether
the model problem is flexural or of membrane–shear. In terms of ( θ̄ , ū , w̄ ) and w1
∼ ∼
and w2 , we define an admissible displacement field v̄ by the formulae (6.3.13). We will
use the notations
= γ ( ū , w̄ ),
γ̄αβ
αβ
∼
ρ̄αβ = ραβ ( θ̄ , ū , w̄ ),
∼ ∼
τ̄α = τα ( θ̄ , ū , w̄ ),
∼ ∼
223
which give the membrane, flexural, and transverse shear strains (6.2.3) engendered by
( θ̄ , ū , w̄ ). By the formulae (6.3.24), we can easily get the expression for χ(v) − χ(v̄),
∼ ∼
which is the difference between the 3D strain tensors engendered by v and v̄:
− γ̄ + t(ρ − ρ̄ ) − t[bλ (γ − γ̄ ) + bλ (γ − γ̄ )]
χαβ (v) − χαβ (v̄) = γαβ
α λβ
αβ
αβ
αβ
λβ
β λα
λα
1
γ ) + bγ (θ − θ̄ )],
− t2 [bα (θγ|β
− θ̄γ|β
γ|α
β γ|α
2
1
χα3 (v) − χα3 (v̄) = χ3α (v) − χ3α (v̄) = (τα − τ̄α ), χ33 (v) − χ33 (v̄) = 0.
2
(7.0.1)
The variant of the model will be justified by proving the convergence rate
kχ(v) − χ(v̄)kE . θ ,
kχ(v)kE (7.0.2)
kχ(v) − χ(v̄)kE = 0,
kχ(v)kE →0
(7.0.3)
or the convergence
lim
which together with the theorems of Chapter 6 give convergence rate or convergence of
the solution of the variant of the model to the 3D solution in the relative energy norm.
7.1
Negligibility of the higher order term in the loading functional
We first show that the higher order term 2 f 1 in the resultant loading func-
tional of the model (6.2.4) is negligible. Let’s just retain the leading term f 0 in the
loading functional, and denote the solution by ( θ̄ , ū , w̄ ) ∈ H. For flexural shells, by
∼ ∼
224
Theorem 3.3.2 and Theorem 3.3.3, we can prove the estimates
k ρ̄ − ρ 0 kLsym (ω) + −1 k γ̄ kLsym (ω) + −1 k τ̄ kL (ω) . K(, ξ 0 , [W ∗ , V ∗ ]) . θ ,
∼
∼ ∼2
∼ ∼2
∼ ∼2
(7.1.1)
if the regularity condition (6.5.5) is satisfied by ξ 0 . If we only have ξ 0 ∈ W ∗ , then
k ρ̄ − ρ 0 kLsym (ω) + −1 k γ̄ kLsym (ω) + −1 k τ̄ kL (ω) → 0 ( → 0).
∼
∼ ∼2
∼ ∼2
∼ ∼2
(7.1.2)
For membrane–shear shells, by Theorem 3.3.4 and Theorem 3.3.5 we have
k ρ̄ kLsym (ω) + k γ̄ − γ 0 kLsym (ω) + k τ̄ − τ 0 kL (ω) . K(, (γ 0 , τ 0 ), [V, W ]) . θ ,
∼
∼
∼
∼
∼
∼ ∼
2
∼2
∼
∼2
(7.1.3)
if the condition (6.6.11) or (6.7.10) is satisfied by (γ 0 , τ 0 ). If we only have (γ 0 , τ 0 ) ∈ V ,
∼
∼
∼
∼
the convergence
k ρ̄ kLsym (ω) + k γ̄ − γ 0 kLsym (ω) + k τ̄ − τ 0 kL (ω) → 0
∼ ∼2
∼
∼ ∼2
∼
∼ ∼2
∼
( → 0)
(7.1.4)
∼
holds. Here, ρ 0 , γ 0 , τ 0 , and ξ 0 are what were defined in Chapter 6. Combining (7.1.1)
∼
∼
∼
with (6.5.6) together with the lower bound (6.5.18), under the condition of Theorem 6.5.2,
we will get the convergence rate (7.0.2) for flexural shells. Combining (7.1.2) with (6.5.4),
under the condition of Theorem 6.5.1, we get the convergence (7.0.3) for flexural shells.
Similarly, under the condition of Theorem 6.6.1 or Theorem 6.7.1, the estimate (7.1.3),
the estimate (6.6.12) or (6.7.11), and the lower bound (6.7.34) together lead to the
convergence rate for membrane–shear shells, and under the condition of Theorem 6.6.1
225
or Theorem 6.7.1, the estimate (7.1.4) and the estimate (6.6.10) or (6.7.9) give the
convergence. Therefore, we just need to keep the leading term f 0 in the resultant
loading functional. Cutting-off the higher order term 2 f 1 will not affect the convergence
property of the model solution to the 3D solution in the relative energy norm.
It should be noted that the higher order term 2 f 1 is also negligible in other norms
for flexural shells and stiff membrane–shear shells, but in the case of the second kind
membrane–shear shells, if f 0 |ker B = 0 but f 1 |ker B 6= 0, although the contribution of
the higher order term 2 f 1 is negligible in the energy norm, but it might be significant in
other norms. To see this, we use ( θ 1 , u 1 , w1 ) to denote the solution of the model (6.2.4)
∼
∼
in which the loading functional is replaced by 2 f 1 . The above analysis has already
shown the negligibility of ( θ 1 , u 1 , w1 ) in the relative energy norm. On the other hand,
∼ ∼
by our analysis of the flexural shell, see (6.5.4), we have the convergence
kρ ( θ 1 , u 1 , w1 ) − ρ 01 kLsym (ω)
∼ ∼ ∼
∼
2
∼
+ −1 kγ ( θ 1 , u 1 , w1 )kLsym (ω) + −1 kτ ( θ 1 , u 1 , w1 )kL (ω) → 0 ( → 0),
∼ ∼ ∼
∼ ∼ ∼
2
∼2
∼
in which ρ 01 6= 0 is defined by the limiting flexural model (6.5.3) with F 0 replaced by
∼
f 1 . So, ( θ 1 , u 1 , w1 ) does not converge to zero in, say, the L2 norm. Therefore, in this
∼ ∼
special case, we can not determine the convergence of the model (6.2.4), either with or
without 2 f 1 , in norms other than the relative energy norm. For plates, asymptotic
analysis shows that the higher order term helps in this case. In the following discussion,
we will discard 2 f 1 . When we mention the model (6.2.4), the loading functional is
understood as f 0 .
226
7.2
The Naghdi model
The Naghdi model can be obtained by replacing the flexural strain operator ρ in
∼
the model (6.2.4) with ρ N . The model reads: Find ( θ , u , w ) ∈ H, such that
∼ ∼
∼
Z
√
1 2
aαβλγ ρN
( θ , u , w )ρN
(φ , y , z) adx
λγ
αβ
∼ ∼
∼ ∼
∼
3
ω
Z
Z
√
√
5
aαβλγ γλγ (u , w )γαβ (y , z) adx + µ
aαβ τβ ( θ , u , w )τα (φ , y , z) adx
+
∼
∼
∼ ∼
∼ ∼
∼ 6
∼
ω
ω
= hf 0 , (φ , y , z)i ∀ (φ , y , z) ∈ H, (7.2.1)
∼ ∼
∼ ∼
in which f 0 is what was defined by (6.2.5), and
1
1
θ , u , w) = (θα|β + θβ|α ) − (bλ
u
+ bλ
ρN
α uλ|β ) + cαβ w.
αβ (∼
∼
2
2 β λ|α
Let’s define ( θ̄ , ū , w̄ ) = ( θ , u , w ). This model can be fitted in the abstract problem
∼ ∼
∼ ∼
of Chapter 3, and classified in the same way in which we classified the model (6.2.4). If
in (7.1.1), (7.1.2), (7.1.3), or (7.1.4), ρ̄ is repalced by ρ N ( θ̄ , ū , w̄ ), these estimates
∼
∼
∼ ∼
hold under exactly the same conditions. From the relation
γ
λ
ραβ = ρN
αβ + bα γλβ + bβ γγα ,
it is easy to see that the estimates (7.1.1), (7.1.2), (7.1.3), or (7.1.4) themselves hold.
By defining the corrections w1 and w2 in the same way as of Chapter 6, we can define
an admissible displacement field v̄ by the formulae (6.3.13). The convergence properties
we established in the last chapter for the model (6.2.4) all apply to this Naghdi model.
227
In the Naghdi model (7.2.1), if the correction factor 5/6 of the transverse shear
term and the factor 5/6 in the first term of the expression of f 0 (6.2.5) are replaced by
1 simultaneously, all the convergence theorems are still true.
7.3
The Koiter model and the Budianski–Sanders model
The Koiter model is defined as the restriction of the Naghdi model (7.2.1) on
the subspace H K = {( θ , u , w) : ( θ , u , w) ∈ H; τ ( θ , u , w) = 0} of H = H 1D (ω) ×
∼ ∼
∼ ∼
∼ ∼ ∼
∼
1 (ω). The constraint τ ( θ , u , w) = 0 is equivalent to θ = −∂ w − bλ u .
H 1D (ω) × HD
α
α
α λ
∼
∼ ∼ ∼
So, it removed the independent variable θ . From this constraint we also see that ∂α w ∈
∼
2 (ω). Constrained on H K , the model
H 1D (ω). Therefore, the space H K = H 1D (ω) × HD
∼
∼
(7.2.1) becomes: Find (u , w ) ∈ H K , such that
∼
Z
√
1 2
(y , z) adx
(u , w )ρK
aαβλγ ρK
αβ
λγ
∼
∼
∼
3
ω
Z
√
+
aαβλγ γλγ (u , w )γαβ (y , z) adx = hf K , (y , z)i ∀ (y , z) ∈ H K . (7.3.1)
∼
ω
∼
∼
∼
∼
The operator ρ K is the restriction of the operator ρ N on H K :
∼
∼
λ
λ
ρK
, w) = −w|αβ − bλ
αβ (u
α|β uλ − (bα uλ|β + bβ uλ|α ) + cαβ w,
∼
γ
2 w − Γ ∂ w, and the resultant loading functional is the restriction of
here, w|αβ = ∂αβ
αβ γ
f 0 on H K :
√
λ
p3o aαβ γαβ (y , z) adx
hf K , (y , z)i = −
∼
∼
∼
2µ + λ ω
Z
228
Z
+
ω
α
3
3 √
α
α γ
[(pα
+
e + qa − 2bγ po )yα + (po |α + pe + qa )z] adx
∼
Z
γT
pα
0 yα ,
The well posedness of this Koiter model easily follows from that of the Naghdi model if we
assume f K is in the dual space of H K . Based on the model solution (u , w ) ∈ H K , we
∼
= −∂ w − bλ u , ū = u , and w̄ = w . By
can define ( θ̄ , ū , w̄ ) ∈ H by setting θ̄α
α
α λ
∼
∼
∼ ∼
defining the transverse corrections w1 and w2 in exactly the same way as of Chapter 6, we
can construct the admissible displacement field v̄ by the formula (6.3.13). The Koiter
model can also be fitted in the abstract problem of Chapter 3 by properly defining
operators and spaces. The problem can be accordingly classified as a flexural shell or
a membrane shell (no shear). For flexural shells, by the same scaling on the loading
functions, the estimate
k ρ̄ − ρ 0 kLsym (ω) + −1 k γ̄ kLsym (ω) . θ , τ̄ = 0,
∼ ∼2
∼
∼
∼ ∼2
(7.3.2)
or the convergence
k ρ̄ − ρ 0 kLsym (ω) + −1 k γ̄ kLsym (ω) → 0 ( → 0), τ̄ = 0
∼
∼ ∼2
∼ ∼2
∼
(7.3.3)
can be proved, depending on the “regularity” of the Lagrange multiplier associated with
a limiting problem which is slightly different from (6.5.3). Here, ρ 0 is the same as what
∼
was defined in (6.5.3). The value of θ might not be the same as what was defined in
(6.5.5). If different, we take the least one to determine the model convergence rate. The
estimate (7.3.2) and the estimate (6.5.6) together will give a convergence rate of the form
229
(7.0.2). The convergence (7.3.3) together with (6.5.4) lead to a convergence of the form
(7.0.3).
For membrane shells, the estimate
k ρ̄ kLsym (ω) + k γ̄ − γ 0 kLsym (ω) . θ ,
∼
∼
∼2
∼
∼2
τ̄ = 0,
∼
(7.3.4)
or the convergence
k ρ̄ kLsym (ω) + k γ̄ − γ 0 kLsym (ω) → 0
∼
∼2
∼
∼
( → 0), τ̄ = 0.
∼
∼2
(7.3.5)
can be proved depending on the “regularity” of γ 0 . Here, γ 0 is what was defined in
∼
∼
(6.6.8) or (6.7.7). Again, the value of θ might be different from that defined in (6.6.11) or
(6.7.10). For example, for a totally clamped elliptic shell, our estimate of the value of θ in
(6.6.11) is 1/6, while from (6.6.28), we know that the value of θ in (7.3.4) should be 1/5.
1 a pβ 6= 0, these estimates are essentially different from (6.6.12)
Note that if τα0 = µ
αβ o
and (6.6.10), or (6.7.11) and (6.7.9). In this case, the difference (7.0.1) can not be small,
so the model can not be justified. Actually, the Koiter model diverges. If, say, pα
o = 0,
then the difference (7.0.1) is small, and the Koiter model can therefore be justified, and
for totally clamped elliptic shells, by (6.6.28), we know that the convergence rate of the
Koiter model is O(1/5 ), which is the same as that of the model (6.2.4).
The Budianski–Sanders model is a variant of the Koiter model. The only difference is in the flexural strain operator. If in the Koiter model (7.3.1), ρ K is replaced by
∼
230
ρ BS that is defined by
∼
1 λ
γ
K
ρBS
αβ = ραβ + 2 (bα γλβ + bβ γγα ),
we will get the Budianski–Sanders model. By using the theory of Chapter 3 and the
above relation, we can easily get a convergence or an estimates of the form (7.3.2),
(7.3.3), (7.3.4), or (7.3.5). So the convergence property of the Budianski–Sanders model
is the same as that of Koiter’s.
7.4
The limiting models
For flexural shells, the limiting model is the variational problem (6.5.3) which is
defined on ker B: Find ( θ 0 , u 0 , w0 ) ∈ ker B, such that
∼
∼
Z
√
1
aαβλγ ρλγ ( θ 0 , u 0 , w0 )ραβ (φ , y , z) adx = hF 0 , (φ , y , z)iH ∗ ×H
∼ ∼
∼ ∼
∼
∼ ∼
3 ω
∀(φ , y , z) ∈ ker B. (7.4.1)
∼ ∼
We let ( θ̄ , ū , w̄ ) = ( θ 0 , u 0 , w0 ), and so we have
∼ ∼
∼
∼
k ρ̄ − ρ 0 kLsym (ω) + −1 k γ̄ kLsym (ω) + −1 k τ̄ kL (ω) = 0.
∼
∼ ∼2
∼ ∼2
∼ ∼2
We define the corrections w1 and w2 with (6.5.8) and (6.5.9). The above equation
together with the estimate (6.5.6) or the convergence (6.5.4) prove the convergence rate
231
of the form (7.0.2) under the condition of Theorem 6.5.2, or the convergence of the form
(7.0.3) under the condition of Theorem 6.5.1.
For totally clamped elliptic shells, we assume that the shell data satisfy the
smoothness assumption of Section 6.6.4. The limiting model reads: Find (u 0 , w0 ) ∈
∼
H01 (ω) × L2 (ω) such that
√
√
λ
aαβλγ γλγ (u 0 , w0 )γαβ (y , z) adx = −
p3o aαβ γαβ (y , z) adx
∼
∼
∼
∼
∼
2µ + λ ω
ω
Z
Z
Z
+
ω
α
α γ
α
3
3 √
[(pα
e + qa − 2bγ po )yα + (po |α + pe + qa )z] adx
∼
∀ (y , z) ∈ H 10 (ω) × L2 (ω). (7.4.2)
∼
∼
By construction, we have already shown in Section 6.6.4 that there exists a ( θ , u , w) ∈ H,
∼ ∼
with u = u 0 , such that
∼
∼
k[γ 0 − γ (u , w), τ 0 − τ ( θ , u , w)]kLsym (ω)×L (ω) + k( θ , u , w)kH . 1/6 .
∼
∼ ∼
∼
∼ ∼ ∼
∼ ∼
2
2
∼
(7.4.3)
∼
Therefore, we can take ( θ̄ , ū , w̄ ) = ( θ , u 0 , w). From (7.4.3), we have
∼ ∼
∼ ∼
k ρ̄ kLsym (ω) + k γ̄ − γ 0 kLsym (ω) + k τ̄ − τ 0 kL (ω) . 1/6 .
∼
∼
∼
∼
∼
2
∼2
∼2
∼
We define w1 and w2 by (6.6.13) and (6.6.14). This estimate and the estimate (6.6.12)
(in which θ = 1/6) together lead to an estimate of the form (7.0.2) in which θ = 1/6.
The convergence rate of v̄ to the 3D solution in the relative energy norm then is O(1/6 ).
232
1
β
If τα0 = aαβ po = 0, we can take ( θ , u , w) ∈ H as defined on page 203, and
∼ ∼
µ
define ( θ̄ , ū , w̄ ) in the same way, we then have
∼ ∼
k ρ̄ kLsym (ω) + k γ̄ − γ 0 kLsym (ω) + k τ̄ − τ 0 kL (ω) . 1/5 .
∼
∼
∼
∼
∼
2
∼2
∼2
∼
The convergence rate of v̄ to the 3D solution in the relative energy norm then is O(1/5 )
Note that without ū = u 0 , we can not say the above argument furnishes a justification
∼
∼
for the limiting model (7.4.2).
For other membrane–shear shells, the limiting model is defined by (6.7.5), the form
of which is the same as the limiting model for totally clamped elliptic shells, but the
]
model is defined on the space VM (ω). It is easy to show that there exists ( θ̄ , ū , w̄ ) ∈ H
∼ ∼
such that an estimate of the form (7.1.3) or (7.1.4) hold, but it seems that the best way
to find such ( θ̄ , ū , w̄ ) might be solving the model (6.2.4). The limiting model is hardly
∼ ∼
useful.
7.5
About the loading assumption
In our analysis of the shell models, we have assumed that the components of the
odd part of surface forces pio , the components of the weighted even part of surface forces
pie , the components of the body force qai , and the coefficients of the rescaled lateral
surface force components pi0 , pi1 , and pi2 are all independent of . This assumption is
different from the assumption assumed in asymptotic theories, see [18]. In this section,
we briefly discuss the justification of the general shell model (6.2.4) under the loading
assumption of asymptotic theories.
233
With a slight abuse of notations, we now use pio to denote the components of
the weighted odd part of the surface forces in this section, i.e., pio = (p̃i+ − p̃i− )/2 .
The meanings of pie , qai , and pi0 , pi1 , and pi2 are the same as before. The new loading
assumption then is that pio , pie , qai , and pi0 , pi1 , and pi2 are all independent of .
With the changed meaning of pio , the form of the resultant loading functional in
the model (6.2.4) will be changed. By replacing pio with pio in (6.2.5) and (6.2.6), the
loading functional will be changed to f 0 + f 1 + 2 f 2 + 3 f 3 , in which, the leading
term is given by
Z
hf 0 , (φ , y , z)i =
∼ ∼
ω
α
3
3 √
+
[(pα
e + qa )yα + (pe + qa )z] adx
∼
Z
γT
pα
0 yα .
This functional is roughly the same as the loading functional obtained by asymptotic
analysis. By using (3.4.9) and (3.4.10) we can get the asymptotic behavior of the model
solution ( θ , u , w ) when → 0. The justification of the model is otherwise the same
∼ ∼
as what we did in Chapter 6. For flexural shells, there is nothing new. For membrane–
shear shells, in the expression of ζ 0 = (γ 0 , τ 0 ), the reformulation of the leading term of
∼
∼
the loading functional (6.7.2), we now have τ 0 = 0. As a consequence, the convergence
∼
(7.1.3) or the estimate (7.1.4) should be replaced by
k ρ̄ kLsym (ω) + k γ̄ − γ 0 kLsym (ω) + k τ̄ kL (ω) . θ
∼
∼
∼
∼
2
∼2
(7.5.1)
∼
∼2
or
k ρ̄ kLsym (ω) + k γ̄ − γ 0 kLsym (ω) + k τ̄ kL (ω) → 0
∼
∼
∼
∼
2
∼2
∼2
∼
( → 0).
(7.5.2)
234
These estimates are similar to (7.3.4) and (7.3.5). Therefore, under the new loading
assumption, when the Naghdi model converges, the Koiter also converges. This is the
reason why under this new loading assumption, there is no significant difference between
the Naghdi-type model and the Koiter-type model.
All the other issues that we discussed in the last few sections can be likewisely
discussed under the new loading assumption. In the literature, the classical models are
usually defined under the loading assumption of this section.
7.6
Concluding remarks
The model was completely justified for plane strain cylindrical shells, flexural
shells, and totally clamped elliptic shells without imposing extra conditions on the shell
data. For other membrane–shear shells, the model was only justified under the assumption that the representation ζ∗0 ∈ W ∗ of the leading term of the resultant loading
functional is in the smaller space V ∗ and the applied forces on the shell is admissible.
A rigorous analysis for the case of ζ∗0 does not belong to V ∗ is completely lacking. By
increasing the number of trial functions in the variational methods, more complicated
models can be derived. It seems that the more involved models might be more accurate
[42] and the range of applicability might also be enlarged.
The mathematical analysis of the derived model given in this thesis is sufficient for
our purpose of proving the model convergence in the relative energy norm, but far from
enough for other purposes. For example, for numerical analysis of the Reissner–Mindlin
plate model, stronger estimates on the model solution are needed and were established
in [7].
235
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Index
P0i , P1i , P2i — coefficients of scaled lateral
A — flexural strain operator, 47, 66, 132,
175
surface force components, 179
Pei — component of scaled weighted even
Aαβλγ — compliance tensor in plane
strain elasticity, 23
part of surface forces, 179
Poi — component of scaled odd part of
Aijkl — compliance tensor, 107
B — maximum absolute value of curva-
surface forces, 179
Peα — component of scaled weighted
ture tensor components, 27, 154
B — membrane–shear strain operator,
even part of surface forces
48, 66, 133, 175
for cylindrical shell, 51
C αβλγ — elasticity tensor in plane
Poα — component of scaled odd part
strain elasticity, 23
of surface forces for cylindrical
C ijkl — elasticity tensor, 107
shell, 51
Qia — component of scaled transverse
E — Young’s modulus, 170
H —mean curvature of middle surface,
average of body force
97
density, 179
Qα
a — component of scaled transverse
H — space on which the model is de-
average of body force density
fined, 47, 66, 120, 151
K — Gauss curvature of middle surface,
for cylindrical shell, 51
Qα
m — component of scaled transverse
97
L — arc length of middle curve, 26
moment of body force density
for cylindrical shell, 51
242
243
R — radius of sphere, 116
Ω — shell body, 26, 98, 150
S — middle curve or middle surface, 26,
F 0 — leading term of scaled resultant
96
U — space containing range of flexural
strain operator, 47, 66, 132, 175
V — closure of W , 47, 66, 133, 176
W — range of membrane–shear strain
operator, 68, 133, 175
Γ∗λ
αβ — Christoffel symbol on cross section of cylinder, 21
Γ∗k
ij — Christoffel symbol, 99
γ
Γαβ — Christoffel symbol on middle surface, 98
Γ0 — left side of the cross section of a
cylinder, 26
ΓL — right side of the cross section of a
cylinder, 26
ΓD — clamping part of shell lateral surface, 105, 150
ΓT — traction part of the lateral surface,
105, 150
Γ± — upper and lower surfaces, 26, 105,
150
loading functional, 51, 179
F 1 — higher order term of scaled resultant loading functional, 51
ai — contravariant basis vector on middle surface, 97
ai — covariant basis vector on middle
surface, 96
aα — covariant basis vector on middle
curve of cylindrical shell cross
section, 26
f 0 — leading term of resultant loading
functional, 34, 122, 153
f 1 — higher order term of resultant
loading functional, 34, 122, 153
g i — contravariant basis vector in the
shell, 98
g i — covariant basis vector in the shell,
98
g α — contravariant basis vector on the
cross section of cylinder, 21
244
g α — covariant basis vector on the cross
section of cylinder, 21
n∗ — unit outer normal on lateral surface, 105
n± — unit outer normals on upper and
lower surfaces of shell, 105
pT — lateral surface force density, 106,
150
p± — upper and lower surface force densities, 106, 150
q — body force density, 106, 150
v — displacement field reconstructed
σ ∗ — stress field determined from the
3D elasticity equations, 108
χαβ — strain tensor in plane strain elasticity, 23
χij — strain tensor, 107
η — ratio of lateral surface area element,
106
γ — middle surface boundary, 96
γ(u, w) — membrane strain on middle
curve of cross section of cylindrical shell, 33
γ — membrane strain engendered by
from the model solution,
cylindrical shell model solution,
128, 155
37
v ∗ — displacement solution of 3D elasticity equations, 108
Φ — mapping defining curvilinear coordinates on shells, 21, 26, 98
φ — mapping defining curvilinear coordinates on middle curve or middle surface, 26, 96
σ — stress field reconstructed from the
model solution, 128, 155
— membrane strain engendered by
γαβ
model solution, 124, 155
γ 0 — limit of membrane strain of cylindrical shell, 53
0 — limit of membrane strain, 135,
γαβ
192, 206
γD — clamping part of the middle surface boundary, 104
245
γT — traction part of the middle surface
boundary, 104
γαβ — membrane strain on middle surface, 121, 152
καβ — reformulation of loading functional, 135, 206
ραβ — flexural strain engendered by the
model solution, 124, 155
ρ0 — flexural strain engendered by the
limiting flexural cylindrical shell
model solution, 52
ρ0αβ — flexural strain engendered by the
λ — Lamé coefficient, 23
limiting flexural model solution,
λ? = 2µλ/(2µ + λ), 33
180
µ — Lamé coefficient, 23
α
α
µα
β = δβ − tbβ , 102
ρN
αβ — Naghdi’s definition of flexural
strain, 122, 153
ν — Poisson ratio, 170
ραβ — flexural strain, 121, 122, 152
ω — coordinate domain of middle sur-
σ ∗αβ — stress solution of plane strain
face, 21, 96
ω — coordinate domain of the shell, 26,
98
ω 1 — scaled coordinate domain of the
shell, 173
elasticity equations, 24
σ ∗ij — stress solution of 3D elasticity
equations, 108
σ011 , σ111 , σ211 , σ012 , σ022 , σ122 —
stress field reconstruction func-
ρ — ratio of volume element, 102
tions for cylindrical shells, 37,
ρ(θ, u, w) — flexural strain in cylindrical
38
shell, 33
ρ — flexural strain engendered by cylindrical shell model solution, 37
αβ αβ αβ
σ0 , σ1 , σ2 , σ03α , σ033 , σ133 —
stress field reconstruction function for spherical shell and general shells, 125, 126, 156, 157
246
τ (θ, u, w) — transverse shear strain in
cylindrical shell, 33
τ — transverse shear strain engendered
by cylindrical shell model solution, 37
τα — transverse shear strain engendered
by model solution, 124, 155
τα0 — limit of transverse shear strain,
135, 192, 206
τα — transverse shear strain, 121, 152
θ — component of cylindrical shell
model solution, 33
— component of model solution, 121,
θα
152
θ0 — component of solution of limiting
cylindrical shell model, 52
0 — component of limiting model soluθα
tion, 179
p̃iT — rescaled lateral surface force component, 112, 118
p̃i± — rescaled upper and lower surface
components, 112, 118, 150
p̃α
± — rescaled upper and lower surface
components in cylindrical shell,
31
q̃ i — rescaled component of body force
density, 112, 118, 151
q̃ α — rescaled component of body force
density in cylindrical shell, 30
ṽi — rescaled displacement components,
113
ṽα — rescaled displacement
component in cylindrical shell,
32
σ̃ αβ — rescaled stress tensor
component in cylindrical shell,
29
σ̃ ij — rescaled stress tensor components,
109
%αβ — constitutive residual for cylindrical shell problem, 41
%ij — constitutive residual, 129, 169
ξ 0 — Lagrange multiplier associated
with limiting flexural shell
model, 73, 180
247
ζ 0 — Riesz representation of reformulated leading term of loading
functional, 52, 74, 78, 84, 135,
192, 206
ζ∗0 — reformulated leading term of load-
dα
β — cofactor of the mixed curvature
tensor, 103
g — determinant of metric tensor of the
shell, 27, 98
g αβ — contravariant metric tensor on
ing functional, 73, 84, 134, 177,
cylindrical shell cross
204
section, 21, 27
ζβα — inverse of µα
β , 103
a — determinant of covariant metric tensor of middle surface, 102
aαβλγ — elasticity tensor of the shell,
121, 152
aαβ — contravariant metric tensor of
middle surface, 97
aαβ — covariant metric tensor of middle
surface, 97
b — curvature of middle curve or sphere,
27, 116
bα
β — mixed curvature tensor of middle
surface, 97
bαβ — covariant curvature tensor of
middle surface, 97
cαβ — the third fundamental form, 97
gαβ — covariant metric tensor on cylindrical shell cross section, 21, 27
g ij — contravariant metric tensor of the
shell, 98
gij — covariant metric tensor of the
shell, 98
n∗i — covariant component of unit outer
normal on lateral surface, 105,
119
nα — covariant components of the unit
outer normal on the boundary of
shell middle surface , 102, 105,
119
pi0 , pi1 , pi2 — coefficients of rescaled lateral surface force density, 120,
151
248
piT — component of lateral surface
force, 106
q i — body force component, 106
qai — component of transverse average of
pia — component of lateral surface
body force density, 151
force average, 120
q α — component of body force density
pim — component of lateral surface
force moment, 120
pie — component of weighted even part
of upper and lower surface
forces, 119, 151
pio — component of odd part of upper
and lower surface
forces, 119, 151
pi± — components of upper and lower
surface forces, 106
pα
e — component of weighted even part
of upper and lower surface
forces for cylindrical shell, 31
pα
o — component of odd part of upper
and lower surface forces for
cylindrical shell, 31
pα
± — component of surface force densi-
in cylindrical shell, 23, 28
qaα — component of transverse average of
body force density in cylindrical
shell, 30
α — component of transverse moment
qm
of body force density in cylindrical shell, 30
r — extra term in the integration identity, 165
u — component of cylindrical shell
model solution, 33, 69
uα — component of model solution, 121,
152
u0 — imiting model solution, 52, 73
u0α — component of limiting model solution, 179, 193
∗ — component of displacement soluvα
ties on upper and lower surfaces
tion of plane strain elasticity
of cylindrical shell, 28
equations, 24
249
vα — component of displacement field
reconstructed from solution of
cylindrical shell model, 22, 41
w — component of model solution, 33,
121, 152
w0 — component of limiting model solution, 52, 179, 193
w1 — transverse deflection correction,
40, 138, 181, 195, 212
w2 — transverse deflection correction,
40, 181, 195, 212